integral
Language: English
Part of speech: noun
Pronunciation: enpr: ĭnʹtĭ-grəl
ipa: /ˈɪntɪɡɹ(ə)l/
enpr: ĭnʹtə-grəl
ipa: /ˈɪntəɡɹəl/
ipa: /ɪnˈtɛɡɹəl/
Forms: form: integrals
Etymology: Borrowed from Middle French integral, from Medieval Latin integrālis, from Latin integer (“entire”); see integer.
Senses: examples: text: The integral of a univariate real-valued function is the area under its curve; but be warned! Not all functions are integrable!
topics: mathematics
sciences
glosses: One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.
topics: mathematics
sciences
glosses: One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.
Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.
examples: text: The integral of #92;frac#123;1#125;#123;x#125; on #91;#92;frac#123;1#125;#123;2#125;,1#93; is #92;ln(2), but the integral of the same function on (0,1#93; diverges. In notation, #92;int#95;#92;frac#123;1#125;#123;2#125;¹#92;frac#123;1#125;#123;x#125;dx#61;#92;ln(2), but #92;int#95;0¹#92;frac#123;1#125;#123;x#125;dx#61;#92;infty.
text: Stokes' theorem relates the integral over a surface of the curl of a vector field to a line integral around the boundary of that surface: #92;int#95;S#92;nabla#92;times#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;S#125;#61;#92;int#95;#123;#92;partialS#125;#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;r#125;,.
topics: mathematics
sciences
glosses: A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.
examples: text: The integral of x² is #92;frac#123;x³#125;#123;3#125; plus a constant.
topics: mathematics
sciences
synonyms: antiderivative
indefinite integral
mul:∫
antonyms: derivative
glosses: An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;
topics: mathematics
sciences
synonyms: antiderivative
indefinite integral
mul:∫
antonyms: derivative
glosses: An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;
The fluent of a given fluxion in Newtonian calculus.
Derived terms: Barnes integral
Choquet integral
contour integral
definite integral
double integral
flux integral
half-integral
Hellinger integral
improper integral
indefinite integral
integral calculus
integral equation
integral theory
integral transform
integration
iterated integral
Jackson integral
Lebesgue integral
line integral
logarithmic integral
Mellin-Barnes integral
ordered integral domain
particular integral
path integral formalism
Riemann integral
Riemann-Stieltjes integral
Sugeno integral
surface integral
triple integral
Volterra integral equation
volume integral
Related terms: integer
integrate