Editing Integral (section)

{{Short description|Operation in mathematical calculus}}
{{About|the concept of definite integrals in calculus|the indefinite integral|antiderivative|the set of numbers|integer|other uses|Integral (disambiguation)}}
{{Redirect|Area under the curve|the pharmacology integral|Area under the curve (pharmacokinetics)|the statistics concept|Receiver operating characteristic#Area under the curve}}
[[File:Integral example.svg|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 <math>f(x)</math> is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example]]
{{Calculus|Integral}}

In [[mathematics]], an '''integral''' is the continuous analog of a [[Summation|sum]], which is used to calculate [[area|areas]], [[volume|volumes]], and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of [[calculus]],<ref group="lower-alpha">Integral calculus is a very well established mathematical discipline for which there are many sources. See {{Harvnb|Apostol|1967}} and {{Harvnb|Anton|Bivens|Davis|2016}}, for example.</ref> the other being [[Derivative|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.

A '''definite integral''' computes the [[signed area]] of the region in the plane that is bounded by the [[Graph of a function|graph]] of a given [[Function (mathematics)|function]] between two points in the [[real line]]. Conventionally, areas above the horizontal [[Coordinate axis|axis]] of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''[[antiderivative]]'', a function whose [[derivative]] is the given function; in this case, they are also called ''indefinite integrals''. The [[fundamental theorem of calculus]] relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are [[inverse function|inverse]] operations.

Although methods of calculating areas and volumes dated from [[ancient Greek mathematics]], the principles of integration were formulated independently by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of [[infinitesimal]] width. [[Bernhard Riemann]] later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a [[Curvilinear coordinates|curvilinear]] region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, [[Henri Lebesgue]] generalized Riemann's formulation by introducing what is now referred to as the [[Lebesgue integration|Lebesgue integral]]; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.

Integrals may be generalized depending on the type of the function as well as the [[Domain (mathematical analysis)|domain]] over which the integration is performed. For example, a [[line integral]] is defined for functions of two or more variables, and the [[Interval (mathematics)|interval]] of integration is replaced by a curve connecting two points in space. In a [[surface integral]], the curve is replaced by a piece of a [[Surface (mathematics)|surface]] in [[three-dimensional space]].