# Elements of the Differential and Integral Calculus/Contents

* CONTENTS   DIFFERENTIAL CALCULUS   CHAPTER I COLLECTION OF FORMULAS
* SECTIONPAGE
* 1.Formulas from Algebra, Trigonometry, and Analytic Geometry................................................................................................................................................................................................................................................................................................................................................................................................1
* 2.Greek alphabet................................................................................................................................................................................................................................................................................................................................................................................................3
* 3.Rules for signs in the four quadrants................................................................................................................................................................................................................................................................................................................................................................................................3
* 4.Natural values of the trigonometric functions................................................................................................................................................................................................................................................................................................................................................................................................4
* 5.Tables of logarithms................................................................................................................................................................................................................................................................................................................................................................................................5
* CHAPTER II VARIABLES AND FUNCTIONS
* 6.Variables and constants................................................................................................................................................................................................................................................................................................................................................................................................6
* 7.Interval of a variable................................................................................................................................................................................................................................................................................................................................................................................................6
* 8.Continuous variation................................................................................................................................................................................................................................................................................................................................................................................................6
* 9.Functions................................................................................................................................................................................................................................................................................................................................................................................................7
* 10.Independent and dependent variables................................................................................................................................................................................................................................................................................................................................................................................................7
* 11.Notation of functions................................................................................................................................................................................................................................................................................................................................................................................................8
* 12.Values of the independent variable for which a function is defined................................................................................................................................................................................................................................................................................................................................................................................................8
* CHAPTER III THEORY OF LIMITS
* 13.Limit of a variable................................................................................................................................................................................................................................................................................................................................................................................................11
* 14.Division by zero excluded................................................................................................................................................................................................................................................................................................................................................................................................12
* 15.Infinitesimals................................................................................................................................................................................................................................................................................................................................................................................................13
* 16.The concept of infinity ($\scriptstyle {\infty }$)................................................................................................................................................................................................................................................................................................................................................................................................13
* 17.Limiting value of a function................................................................................................................................................................................................................................................................................................................................................................................................14
* 18.Continuous and discontinuous functions................................................................................................................................................................................................................................................................................................................................................................................................14
* 19.Continuity and discontinuity of functions illustrated by their graphs................................................................................................................................................................................................................................................................................................................................................................................................16
* 20.Fundamental theorems on limits................................................................................................................................................................................................................................................................................................................................................................................................18
* 21.Special limiting values................................................................................................................................................................................................................................................................................................................................................................................................20
* 22.The limit of $\scriptstyle {\frac {\sin x}{x}}$ as $\scriptstyle {x\doteq 0}$................................................................................................................................................................................................................................................................................................................................................................................................21
* 23.The number $\scriptstyle {e}$................................................................................................................................................................................................................................................................................................................................................................................................22
* 24.Expressions assuming the form $\scriptstyle {\frac {\infty }{\infty }}$................................................................................................................................................................................................................................................................................................................................................................................................23
* CHAPTER IV DIFFERENTIATION
* 25.Introduction................................................................................................................................................................................................................................................................................................................................................................................................25
* 26.Increments................................................................................................................................................................................................................................................................................................................................................................................................25
* 27.Comparison of increments................................................................................................................................................................................................................................................................................................................................................................................................26
* 28.Derivative of a function of one variable................................................................................................................................................................................................................................................................................................................................................................................................27
* 29.Symbols for derivatives................................................................................................................................................................................................................................................................................................................................................................................................28
* 30.Differentiable functions................................................................................................................................................................................................................................................................................................................................................................................................29
* 31.General rule for differentiation................................................................................................................................................................................................................................................................................................................................................................................................29
* 32.Applications of the derivative to Geometry................................................................................................................................................................................................................................................................................................................................................................................................31
* CHAPTER V RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS
* 33.Importance of General Rule................................................................................................................................................................................................................................................................................................................................................................................................34
* 34.Differentiation of a constant................................................................................................................................................................................................................................................................................................................................................................................................36
* 35.Differentiation of a variable with respect to itself................................................................................................................................................................................................................................................................................................................................................................................................37
* 36.Differentiation of a sum................................................................................................................................................................................................................................................................................................................................................................................................37
* 37.Differentiation of the product of a constant and a function................................................................................................................................................................................................................................................................................................................................................................................................37
* 38.Differentiation of the product of two functions................................................................................................................................................................................................................................................................................................................................................................................................38
* 39.Differentiation of the product of any finite number of functions................................................................................................................................................................................................................................................................................................................................................................................................38
* 40.Differentiation of a function with a constant exponent................................................................................................................................................................................................................................................................................................................................................................................................39
* 41.Differentiation of a quotient................................................................................................................................................................................................................................................................................................................................................................................................40
* 42.Differentiation of a function of a function................................................................................................................................................................................................................................................................................................................................................................................................44
* 43.Differentiation of inverse functions................................................................................................................................................................................................................................................................................................................................................................................................45
* 44.Differentiation of a logarithm................................................................................................................................................................................................................................................................................................................................................................................................46
* 45.Differentiation of the simple exponential function................................................................................................................................................................................................................................................................................................................................................................................................48
* 46.Differentiation of the general exponential function................................................................................................................................................................................................................................................................................................................................................................................................49
* 47.Logarithmic differentiation................................................................................................................................................................................................................................................................................................................................................................................................50
* 48.Differentiation of $\scriptstyle {\sin v}$................................................................................................................................................................................................................................................................................................................................................................................................54
* 49.Differentiation of $\scriptstyle {\cos v}$................................................................................................................................................................................................................................................................................................................................................................................................55
* 50.Differentiation of $\scriptstyle {\tan v}$................................................................................................................................................................................................................................................................................................................................................................................................56
* 51.Differentiation of $\scriptstyle {\cot v}$................................................................................................................................................................................................................................................................................................................................................................................................56
* 52.Differentiation of $\scriptstyle {\sec v}$................................................................................................................................................................................................................................................................................................................................................................................................56
* 53.Differentiation of $\scriptstyle {\csc v}$................................................................................................................................................................................................................................................................................................................................................................................................57
* 54.Differentiation of $\scriptstyle {\operatorname {vers} v}$................................................................................................................................................................................................................................................................................................................................................................................................57
* 55.Differentiation of $\scriptstyle {\operatorname {arc~sin} v}$................................................................................................................................................................................................................................................................................................................................................................................................61
* 56.Differentiation of $\scriptstyle {\operatorname {arc~cos} v}$................................................................................................................................................................................................................................................................................................................................................................................................62
* 57.Differentiation of $\scriptstyle {\operatorname {arc~tan} v}$................................................................................................................................................................................................................................................................................................................................................................................................62
* 58.Differentiation of $\scriptstyle {\operatorname {arc~cot} v}$................................................................................................................................................................................................................................................................................................................................................................................................63
* 59.Differentiation of $\scriptstyle {\operatorname {arc~sec} v}$................................................................................................................................................................................................................................................................................................................................................................................................63
* 60.Differentiation of $\scriptstyle {\operatorname {arc~csc} v}$................................................................................................................................................................................................................................................................................................................................................................................................64
* 61.Differentiation of $\scriptstyle {\operatorname {arc~vers} v}$................................................................................................................................................................................................................................................................................................................................................................................................65
* 62.Implicit functions................................................................................................................................................................................................................................................................................................................................................................................................69
* 63.Differentiation of implicit functions................................................................................................................................................................................................................................................................................................................................................................................................69
* CHAPTER VI SIMPLE APPLICATIONS OF THE DERIVATIVE
* 64.Direction of a curve................................................................................................................................................................................................................................................................................................................................................................................................73
* 65.Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates................................................................................................................................................................................................................................................................................................................................................................................................76
* 66.Parametric equations of a curve................................................................................................................................................................................................................................................................................................................................................................................................79
* 67.Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point................................................................................................................................................................................................................................................................................................................................................................................................83
* 68.Lengths of polar subtangent and polar subnormal................................................................................................................................................................................................................................................................................................................................................................................................86
* 69.Solution of equations having multiple roots................................................................................................................................................................................................................................................................................................................................................................................................88
* 70.Applications of the derivative in mechanics. Velocity................................................................................................................................................................................................................................................................................................................................................................................................90
* 71.Component velocities................................................................................................................................................................................................................................................................................................................................................................................................91
* 72.Acceleration................................................................................................................................................................................................................................................................................................................................................................................................92
* 73.Component accelerations................................................................................................................................................................................................................................................................................................................................................................................................93
* CHAPTER VII SUCCESSIVE DIFFERENTIATION
* 74.Definition of successive derivatives................................................................................................................................................................................................................................................................................................................................................................................................97
* 75.Notation................................................................................................................................................................................................................................................................................................................................................................................................97
* 76.The $\scriptstyle {n{\text{th}}}$ derivative................................................................................................................................................................................................................................................................................................................................................................................................98
* 77.Leibnitz's formula for the $\scriptstyle {n{\text{th}}}$ derivative of a product................................................................................................................................................................................................................................................................................................................................................................................................98
* 78.Successive differentiation of implicit functions................................................................................................................................................................................................................................................................................................................................................................................................100
* CHAPTER VIII MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING
* 79.Introduction................................................................................................................................................................................................................................................................................................................................................................................................103
* 80.Increasing and decreasing functions................................................................................................................................................................................................................................................................................................................................................................................................106
* 81.Tests for determining when a function is increasing and when decreasing................................................................................................................................................................................................................................................................................................................................................................................................108
* 82.Maximum and minimum values of a function................................................................................................................................................................................................................................................................................................................................................................................................109
* 83.First method for examining a function for maximum and minimum values................................................................................................................................................................................................................................................................................................................................................................................................111
* 84.Second method for examining a function for maximum and minimum values................................................................................................................................................................................................................................................................................................................................................................................................112
* 85.Definition of points of inflection and rule for finding points of inflection................................................................................................................................................................................................................................................................................................................................................................................................125
* 86.Curve tracing................................................................................................................................................................................................................................................................................................................................................................................................128
* CHAPTER IX DIFFERENTIALS
* 87.Introduction................................................................................................................................................................................................................................................................................................................................................................................................131
* 88.Definitions................................................................................................................................................................................................................................................................................................................................................................................................131
* 89.Infinitesimals................................................................................................................................................................................................................................................................................................................................................................................................132
* 90.Derivative of the arc in rectangular coördinates................................................................................................................................................................................................................................................................................................................................................................................................134
* 91.Derivative of the arc in polar coördinates................................................................................................................................................................................................................................................................................................................................................................................................135
* 92.Formulas for finding the differentials of functions................................................................................................................................................................................................................................................................................................................................................................................................137
* 93.Successive differentials................................................................................................................................................................................................................................................................................................................................................................................................139
* CHAPTER X RATES
* 94.The derivative considered as the ratio of two rates................................................................................................................................................................................................................................................................................................................................................................................................141
* CHAPTER XI CHANGE OF VARIABLE
* 95.Interchange of dependent and independent variables................................................................................................................................................................................................................................................................................................................................................................................................148
* 96.Change of the dependent variable................................................................................................................................................................................................................................................................................................................................................................................................149
* 97.Change of the independent variable................................................................................................................................................................................................................................................................................................................................................................................................150
* 98.Simultaneous change of both independent and dependent variables................................................................................................................................................................................................................................................................................................................................................................................................152
* CHAPTER XII CURVATURE. RADIUS OF CURVATURE
* 99.Curvature................................................................................................................................................................................................................................................................................................................................................................................................155
* 100.Curvature of a circle................................................................................................................................................................................................................................................................................................................................................................................................155
* 101.Curvature at a point................................................................................................................................................................................................................................................................................................................................................................................................156
* 102.Formulas for curvature................................................................................................................................................................................................................................................................................................................................................................................................159
* 103.Radius of curvature................................................................................................................................................................................................................................................................................................................................................................................................159
* 104.Circle of curvature................................................................................................................................................................................................................................................................................................................................................................................................161
* CHAPTER XIII THEOREM OF MEAN VALUE. INDETERMINATE FORMS
* 105.Rolle's Theorem................................................................................................................................................................................................................................................................................................................................................................................................164
* 106.The Theorem of Mean Value................................................................................................................................................................................................................................................................................................................................................................................................165
* 107.The Extended Theorem of Mean Value................................................................................................................................................................................................................................................................................................................................................................................................166
* 108.Maxima and minima treated analytically................................................................................................................................................................................................................................................................................................................................................................................................167
* 109.Indeterminate forms................................................................................................................................................................................................................................................................................................................................................................................................170
* 110.Evaluation of a function taking on an indeterminate form................................................................................................................................................................................................................................................................................................................................................................................................170
* 111.Evaluation of the indeterminate form $\scriptstyle {\frac {0}{0}}$................................................................................................................................................................................................................................................................................................................................................................................................171
* 112.Evaluation of the indeterminate form $\scriptstyle {\frac {\infty }{\infty }}$................................................................................................................................................................................................................................................................................................................................................................................................174
* 113.Evaluation of the indeterminate form $\scriptstyle {0\cdot \infty }$................................................................................................................................................................................................................................................................................................................................................................................................174
* 114.Evaluation of the indeterminate form $\scriptstyle {\infty -\infty }$................................................................................................................................................................................................................................................................................................................................................................................................175
* 115.Evaluation of the indeterminate forms $\scriptstyle {0^{0},~1^{\infty },~\infty ^{0}}$................................................................................................................................................................................................................................................................................................................................................................................................176
* CHAPTER XIV CIRCLE OF CURVATURE. CENTER OF CURVATURE
* 116.Circle of curvature. Center of curvature................................................................................................................................................................................................................................................................................................................................................................................................178
* 117.Second method for finding center of curvature................................................................................................................................................................................................................................................................................................................................................................................................180
* 118.Center of curvature the limiting position of the intersection of normals at neighboring points................................................................................................................................................................................................................................................................................................................................................................................................181
* 119.Evolutes................................................................................................................................................................................................................................................................................................................................................................................................182
* 120.Properties of the evolute................................................................................................................................................................................................................................................................................................................................................................................................186
* 121.Involutes and their mechanical construction................................................................................................................................................................................................................................................................................................................................................................................................187
* CHAPTER XV PARTIAL DIFFERENTIATION
* 122.Continuous functions of two or more independent variables................................................................................................................................................................................................................................................................................................................................................................................................190
* 123.Partial derivatives................................................................................................................................................................................................................................................................................................................................................................................................191
* 124.Partial derivatives interpreted geometrically................................................................................................................................................................................................................................................................................................................................................................................................192
* 125.Total derivatives................................................................................................................................................................................................................................................................................................................................................................................................194
* 126.Total differentials................................................................................................................................................................................................................................................................................................................................................................................................197
* 127.Differentiation of implicit functions................................................................................................................................................................................................................................................................................................................................................................................................198
* 128.Successive partial derivatives................................................................................................................................................................................................................................................................................................................................................................................................202
* 129.Order of differentiation immaterial................................................................................................................................................................................................................................................................................................................................................................................................203
* CHAPTER XVI ENVELOPES
* 130.Family of curves. Variable parameter................................................................................................................................................................................................................................................................................................................................................................................................205
* 131.Envelope of a family of curves depending on one parameter................................................................................................................................................................................................................................................................................................................................................................................................205
* 132.The evolute of a given curve considered as the envelope of its normals................................................................................................................................................................................................................................................................................................................................................................................................208
* 133.Two parameters connected by one equation of condition................................................................................................................................................................................................................................................................................................................................................................................................209
* CHAPTER XVII SERIES
* 134.Introduction................................................................................................................................................................................................................................................................................................................................................................................................212
* 135.Infinite series................................................................................................................................................................................................................................................................................................................................................................................................213
* 136.Existence of a limit................................................................................................................................................................................................................................................................................................................................................................................................215
* 137.Fundamental test for convergence................................................................................................................................................................................................................................................................................................................................................................................................216
* 138.Comparison test for convergence................................................................................................................................................................................................................................................................................................................................................................................................217
* 139.Cauchy's ratio test for convergence................................................................................................................................................................................................................................................................................................................................................................................................218
* 140.Alternating series................................................................................................................................................................................................................................................................................................................................................................................................220
* 141.Absolute convergence................................................................................................................................................................................................................................................................................................................................................................................................220
* 142.Power series................................................................................................................................................................................................................................................................................................................................................................................................223
* CHAPTER XVIII EXPANSION OF FUNCTIONS
* 143.Introduction................................................................................................................................................................................................................................................................................................................................................................................................227
* 144.Taylor's Theorem and Taylor's Series................................................................................................................................................................................................................................................................................................................................................................................................228
* 145.Maclaurin's Theorem and Maclaurin's Series................................................................................................................................................................................................................................................................................................................................................................................................230
* 146.Computation by series................................................................................................................................................................................................................................................................................................................................................................................................234
* 147.Approximate formulas derived from series. Interpolation................................................................................................................................................................................................................................................................................................................................................................................................237
* 148.Taylor's Theorem for functions of two or more variables................................................................................................................................................................................................................................................................................................................................................................................................240
* 149.Maxima and minima of functions of two independent variables................................................................................................................................................................................................................................................................................................................................................................................................243
* CHAPTER XIX ASYMPTOTES. SINGULAR POINTS
* 150.Rectilinear asymptotes................................................................................................................................................................................................................................................................................................................................................................................................249
* 151.Asymptotes found by method of limiting intercepts................................................................................................................................................................................................................................................................................................................................................................................................249
* 152.Method of determining asymptotes to algebraic curves................................................................................................................................................................................................................................................................................................................................................................................................250
* 153.Asymptotes in polar coördinates................................................................................................................................................................................................................................................................................................................................................................................................254
* 154.Singular points................................................................................................................................................................................................................................................................................................................................................................................................255
* 155.Determination of the tangent to an algebraic curve at a given point by inspection................................................................................................................................................................................................................................................................................................................................................................................................255
* 156.Nodes................................................................................................................................................................................................................................................................................................................................................................................................258
* 157.Cusps................................................................................................................................................................................................................................................................................................................................................................................................259
* 158.Conjugate or isolated points................................................................................................................................................................................................................................................................................................................................................................................................260
* 159.Transcendental singularities................................................................................................................................................................................................................................................................................................................................................................................................260
* CHAPTER XX APPLICATIONS TO GEOMETRY OF SPACE
* 160.Tangent line and normal plane to a skew curve whose equations are given in parametric form................................................................................................................................................................................................................................................................................................................................................................................................262
* 161.Tangent plane to a surface................................................................................................................................................................................................................................................................................................................................................................................................264
* 162.Normal line to a surface................................................................................................................................................................................................................................................................................................................................................................................................266
* 163.Another form of the equations of the tangent line to a skew curve................................................................................................................................................................................................................................................................................................................................................................................................268
* 164.Another form of the equation of the normal plane to a skew curve................................................................................................................................................................................................................................................................................................................................................................................................269
* CHAPTER XXI CURVES FOR REFERENCE     INTEGRAL CALCULUS   CHAPTER XXII INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS
* 165.Integration................................................................................................................................................................................................................................................................................................................................................................................................279
* 166.Constant of integration. Indefinite integral................................................................................................................................................................................................................................................................................................................................................................................................281
* 167.Rules for integrating standard elementary forms................................................................................................................................................................................................................................................................................................................................................................................................282
* 168.Trigonometric differentials................................................................................................................................................................................................................................................................................................................................................................................................298
* 169.Integration of expressions containing $\scriptstyle {\sqrt {a^{2}-x^{2}}}$ or $\scriptstyle {\sqrt {x^{2}\pm a^{2}}}$ by a trigonometric substitution................................................................................................................................................................................................................................................................................................................................................................................................304
* CHAPTER XXIII CONSTANT OF INTEGRATION
* 170.Determination of the constant of integration by means of initial conditions................................................................................................................................................................................................................................................................................................................................................................................................307
* 171.Geometrical signification of the constant of integration................................................................................................................................................................................................................................................................................................................................................................................................307
* 172.Physical signification of the constant of integration................................................................................................................................................................................................................................................................................................................................................................................................309
* CHAPTER XXIV THE DEFINITE INTEGRAL
* 173.Differential of an area................................................................................................................................................................................................................................................................................................................................................................................................314
* 174.The definite integral................................................................................................................................................................................................................................................................................................................................................................................................314
* 175.Calculation of a definite integral................................................................................................................................................................................................................................................................................................................................................................................................316
* 176.Calculation of areas................................................................................................................................................................................................................................................................................................................................................................................................318
* 177.Geometrical representation of an integral................................................................................................................................................................................................................................................................................................................................................................................................319
* 178.Mean value of $\scriptstyle {\phi (x)}$................................................................................................................................................................................................................................................................................................................................................................................................320
* 179.Interchange of limits................................................................................................................................................................................................................................................................................................................................................................................................320
* 180.Decomposition of the interval................................................................................................................................................................................................................................................................................................................................................................................................321
* 181.The definite integral a function of its limits................................................................................................................................................................................................................................................................................................................................................................................................321
* 182.Infinite limits................................................................................................................................................................................................................................................................................................................................................................................................321
* 183.When $\scriptstyle {y=\phi (x)}$ is discontinuous................................................................................................................................................................................................................................................................................................................................................................................................322
* CHAPTER XXV INTEGRATION OF RATIONAL FRACTIONS
* 184.Introduction................................................................................................................................................................................................................................................................................................................................................................................................325
* 185.Case I................................................................................................................................................................................................................................................................................................................................................................................................325
* 186.Case II................................................................................................................................................................................................................................................................................................................................................................................................327
* 187.Case III................................................................................................................................................................................................................................................................................................................................................................................................329
* 188.Case IV................................................................................................................................................................................................................................................................................................................................................................................................331
* CHAPTER XXVI INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION
* 189.Introduction................................................................................................................................................................................................................................................................................................................................................................................................335
* 190.Differentials containing fractional powers of $\scriptstyle {x}$ only................................................................................................................................................................................................................................................................................................................................................................................................335
* 191.Differentials containing fractional powers of $\scriptstyle {a+bx}$ only................................................................................................................................................................................................................................................................................................................................................................................................336
* 192.Change in limits corresponding to change in variable................................................................................................................................................................................................................................................................................................................................................................................................336
* 193.Differentials containing no radical except $\scriptstyle {\sqrt {a+bx+x^{2}}}$................................................................................................................................................................................................................................................................................................................................................................................................338
* 194.Differentials containing no radical except $\scriptstyle {\sqrt {a+bx-x^{2}}}$................................................................................................................................................................................................................................................................................................................................................................................................338
* 195.Binomial differentials................................................................................................................................................................................................................................................................................................................................................................................................340
* 196.Conditions of integrability of binomial differentials................................................................................................................................................................................................................................................................................................................................................................................................341
* 197.Transformation of trigonometric differentials................................................................................................................................................................................................................................................................................................................................................................................................343
* 198.Miscellaneous substitutions................................................................................................................................................................................................................................................................................................................................................................................................345
* CHAPTER XXVII INTEGRATION BY PARTS. REDUCTION FORMULAS
* 199.Formula for integration by parts................................................................................................................................................................................................................................................................................................................................................................................................347
* 200.Reduction formulas for binomial differentials................................................................................................................................................................................................................................................................................................................................................................................................350
* 201.Reduction formulas for trigonometric differentials................................................................................................................................................................................................................................................................................................................................................................................................356
* 202.To find $\scriptstyle {\int e^{ax}\sin {nx}dx}$ and $\scriptstyle {\int e^{ax}\cos {nx}dx}$................................................................................................................................................................................................................................................................................................................................................................................................359
* CHAPTER XXVIII INTEGRATION A PROCESS OF SUMMATION
* 203.Introduction................................................................................................................................................................................................................................................................................................................................................................................................361
* 204.The fundamental theorem of Integral Calculus................................................................................................................................................................................................................................................................................................................................................................................................361
* 205.Analytical proof of the Fundamental Theorem................................................................................................................................................................................................................................................................................................................................................................................................364
* 206.Areas of plane curves. Rectangular coördinates................................................................................................................................................................................................................................................................................................................................................................................................365
* 207.Area when curve is given in parametric form................................................................................................................................................................................................................................................................................................................................................................................................368
* 208.Areas of plane curves. Polar coördinates................................................................................................................................................................................................................................................................................................................................................................................................370
* 209.Length of a curve................................................................................................................................................................................................................................................................................................................................................................................................372
* 210.Lengths of plane curves. Rectangular coördinates................................................................................................................................................................................................................................................................................................................................................................................................373
* 211.Lengths of plane curves. Polar coördinates................................................................................................................................................................................................................................................................................................................................................................................................375
* 212.Volumes of solids of revolution................................................................................................................................................................................................................................................................................................................................................................................................377
* 213.Areas of surfaces of revolution................................................................................................................................................................................................................................................................................................................................................................................................381
* 214.Miscellaneous applications................................................................................................................................................................................................................................................................................................................................................................................................385
* CHAPTER XXIX SUCCESSIVE AND PARTIAL INTEGRATION
* 215.Successive integration................................................................................................................................................................................................................................................................................................................................................................................................393
* 216.Partial integration................................................................................................................................................................................................................................................................................................................................................................................................395
* 217.Definite double integral. Geometric interpretation................................................................................................................................................................................................................................................................................................................................................................................................396
* 218.Value of a definite double integral over a region................................................................................................................................................................................................................................................................................................................................................................................................400
* 219.Plane area as a definite double integral. Rectangular coördinates................................................................................................................................................................................................................................................................................................................................................................................................402
* 220.Plane area as a definite double integral. Polar coördinates................................................................................................................................................................................................................................................................................................................................................................................................406
* 221.Moment of area................................................................................................................................................................................................................................................................................................................................................................................................408
* 222.Center of area................................................................................................................................................................................................................................................................................................................................................................................................408
* 223.Moment of inertia. Plane areas................................................................................................................................................................................................................................................................................................................................................................................................410
* 224.Polar moment of inertia. Rectangular coördinates................................................................................................................................................................................................................................................................................................................................................................................................410
* 225.Polar moment of inertia. Polar coördinates................................................................................................................................................................................................................................................................................................................................................................................................411
* 226.General method for finding the areas of surfaces................................................................................................................................................................................................................................................................................................................................................................................................413
* 227.Volumes found by triple integration................................................................................................................................................................................................................................................................................................................................................................................................417
* CHAPTER XXX ORDINARY DIFFERENTIAL EQUATIONS
* 228.Differential equations. Order and degree................................................................................................................................................................................................................................................................................................................................................................................................421
* 229.Solutions of differential equations................................................................................................................................................................................................................................................................................................................................................................................................422
* 230.Verifications of solutions................................................................................................................................................................................................................................................................................................................................................................................................423
* 231.Differential equations of the first order and of the first degree................................................................................................................................................................................................................................................................................................................................................................................................424
* 232.Differential equations of the $\scriptstyle {n{\text{th}}}$ order and of the first degree................................................................................................................................................................................................................................................................................................................................................................................................432
* CHAPTER XXXI INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS
* 233.Mechanical integration................................................................................................................................................................................................................................................................................................................................................................................................443
* 234.Integral curves................................................................................................................................................................................................................................................................................................................................................................................................443
* 235.The integraph................................................................................................................................................................................................................................................................................................................................................................................................445
* 236.Polar planimeter................................................................................................................................................................................................................................................................................................................................................................................................446
* 237.Area swept over by a line................................................................................................................................................................................................................................................................................................................................................................................................446
* 238.Approximate integration................................................................................................................................................................................................................................................................................................................................................................................................448
* 239.Trapezoidal rule................................................................................................................................................................................................................................................................................................................................................................................................448
* 240.Simpson's rule (parabolic rule)................................................................................................................................................................................................................................................................................................................................................................................................449
* 241.Integrals for reference................................................................................................................................................................................................................................................................................................................................................................................................451
* INDEX................................................................................................................................................................................................................................................................................................................................................................................................461