integral

Also known as integral of x, integral of a function, integral of f(x) with respect to the variable x

thumb|300px|A definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of f(x) between a and b is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example

AI overview

An integral is a mathematical tool that calculates the area under a curve on a graph, accounting for whether regions are above or below the horizontal axis. It matters because it allows mathematicians and scientists to find total quantities like distance, area, and volume from information about how things change.

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Wikidata facts

Image: Riemann sum convergence.png

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Commons category: Integration (mathematics)
Stack Exchange tag: math.stackexchange.com/tags/integration
TeX string: \int
Unicode character: ∫

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~45 min read

Article

43 sections

Contents

History
Pre-calculus integration
Leibniz and Newton
Formalization
Historical notation
First use of the term
Terminology and notation
Interpretations
Formal definitions
Riemann integral
Lebesgue integral
Other integrals
Properties
Linearity
Inequalities
Conventions
Fundamental theorem of calculus
First theorem
Second theorem
Extensions
Improper integrals
Multiple integration
Line integrals and surface integrals
Contour integrals
Integrals of differential forms
Summations
Functional integrals
Applications
Computation
Analytical
Symbolic
Numerical
Mechanical
Geometrical
Integration by differentiation
Examples
Using the fundamental theorem of calculus
See also
Notes
References
Bibliography
External links
Online books

In mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral, called integration, is one of the two fundamental operations of calculus, along with differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.