# Disc integration

Part of a series of articles about :
Calculus :
$\int _{a}^{b}f'(t)\,dt=f(b)-f(a)$ :
Fundamental theorem
Limits
Continuity
Rolle's theorem
Mean value theorem
Inverse function theorem :
Differential
Definitions
Derivative (generalizations)
Differential
infinitesimal
of a function
total
Concepts
Differentiation notation
Second derivative
Implicit differentiation
Logarithmic differentiation
Related rates
Taylor's theorem
Rules and identities
Sum
Product
Chain
Power
Quotient
L'Hôpital's rule
Inverse
General Leibniz
Faà di Bruno's formula
Reynolds :
Integral
Lists of integrals
Integral transform
Leibniz integral rule
Definitions
Antiderivative
Integral (improper)
Riemann integral
Lebesgue integration
Contour integration
Integral of inverse functions
Integration by
Parts
Discs
Cylindrical shells
Substitution (trigonometric, tangent half-angle, Euler)
Euler's formula
Partial fractions (Heaviside's method)
Changing order
Reduction formulae
Differentiating under the integral sign
Risch algorithm :
Series
Geometric (arithmetico-geometric)
Harmonic
Alternating
Power
Binomial
Taylor
Convergence tests
Summand limit (term test)
Ratio
Root
Integral
Direct comparison
Limit comparison
Alternating series
Cauchy condensation
Dirichlet
Abel :
Vector
Gradient
Divergence
Curl
Laplacian
Directional derivative
Identities
Theorems
Gradient
Green's
Stokes'
Divergence
generalized Stokes
Helmholtz decomposition :
Multivariable
Formalisms
Matrix
Tensor
Exterior
Geometric
Definitions
Partial derivative
Multiple integral
Line integral
Surface integral
Volume integral
Jacobian
Hessian :
Advanced
Calculus on Euclidean space
Generalized functions
Limit of distributions :
Specialized
Fractional
Malliavin
Stochastic
Variations :
Miscellanea
Precalculus
History
Glossary
List of topics
Integration Bee
Mathematical analysis
Nonstandard analysis :
vte :
Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs (the "washer method") to obtain hollow solids of revolutions. This is in contrast to shell integration, that integrates along an axis perpendicular to the axis of revolution.

## Definition
### Function of x

If the function to be revolved is a function of x, the following integral represents the volume of the solid of revolution:
    $\pi \int _{a}^{b}R(x)^{2}\,dx$
where R(x) is the distance between the function and the axis of rotation. This works only if the axis of rotation is horizontal (example: y = 3 or some other constant).

### Function of y

If the function to be revolved is a function of y, the following integral will obtain the volume of the solid of revolution:
    $\pi \int _{c}^{d}R(y)^{2}\,dy$
where R(y) is the distance between the function and the axis of rotation. This works only if the axis of rotation is vertical (example: x = 4 or some other constant).

### Washer method

To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. This can be calculated in a single integral similar to the following:
    $\pi \int _{a}^{b}\left(R_{\mathrm {O} }(x)^{2}-R_{\mathrm {I} }(x)^{2}\right)\,dx$
where R$_O$(x) is the function that is furthest from the axis of rotation and R$_I$(x) is the function that is closest to the axis of rotation. For example, the next figure shows the rotation along the x-axis of the red "leaf" enclosed between the square-root and quadratic curves:
The volume of this solid is:
    $\pi \int _{0}^{1}\left(\left({\sqrt {x}}\right)^{2}-\left(x^{2}\right)^{2}\right)\,dx\,.$
One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.
    $R_{\mathrm {O} }(x)^{2}-R_{\mathrm {I} }(x)^{2}\neq \left(R_{\mathrm {O} }(x)-R_{\mathrm {I} }(x)\right)^{2}$
(This formula only works for revolutions about the x-axis.)
To rotate about any horizontal axis, simply subtract from that axis from each formula. If h is the value of a horizontal axis, then the volume equals
    $\pi \int _{a}^{b}\left(\left(h-R_{\mathrm {O} }(x)\right)^{2}-\left(h-R_{\mathrm {I} }(x)\right)^{2}\right)\,dx\,.$
For example, to rotate the region between y = −2x + x$^2$ and y = x along the axis y = 4, one would integrate as follows:
    $\pi \int _{0}^{3}\left(\left(4-\left(-2x+x^{2}\right)\right)^{2}-(4-x)^{2}\right)\,dx\,.$
The bounds of integration are the zeros of the first equation minus the second. Note that when integrating along an axis other than the x, the graph of the function which is furthest from the axis of rotation may not be obvious. In the previous example, even though the graph of y = x is, with respect to the x-axis, further up than the graph of y = −2x + x$^2$, with respect to the axis of rotation the function y = x is the inner function: its graph is closer to y = 4 or the equation of the axis of rotation in the example.
The same idea can be applied to both the y-axis and any other vertical axis. One simply must solve each equation for x before one inserts them into the integration formula.