integral
nounUK /ˈɪntɪɡɹ(ə)l/
Etymology Borrowed from Middle French integral, from Medieval Latin integrālis, from Latin integer (“entire”); see integer.
Mathematics, Sciences 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.
- The integral of a univariate real-valued function is the area under its curve; but be warned! Not all functions are integrable!
Mathematics, Sciences Specifically Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.
Mathematics, Sciences 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.
- 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.
- 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;,.
Mathematics, Sciences 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 integral of x² is #92;frac#123;x³#125;#123;3#125; plus a constant.
Mathematics, Sciences Historical, Obsolete The fluent of a given fluxion in Newtonian calculus.
- Derived terms
31 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