In mathematicsan integral is the continuous analog of a sumwhich is used to calculate areasvolumesand their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus[a] the other being differentiation.
Integration started as a method to solve problems in mathematics and physicssuch as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields. The integrals enumerated here are called definite integralswhich can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line.
Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivativea function whose derivative is the given function; in this case, they are also called indefinite integrals.
The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations. Although methods of calculating areas and volumes dated from ancient Greek mathematicsthe 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 integral matte engelska a 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 integral ; it is more robust 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 over which the integration is performed. For example, a line integral integral matte engelska defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval.
In a surface integralthe curve is replaced by a piece of a surface in three-dimensional space. The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus ca.
A similar method was independently developed in China around the 3rd century AD by Liu Huiwho used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere.
The next significant advances in integral calculus did not begin to appear until the 17th century. Further steps were made in the early 17th century by Barrow and Torricelliwho provided the first hints of a connection between integration and differentiation.
Barrow provided the first proof of the fundamental theorem of calculus. The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals.
In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains.
This framework eventually became modern calculuswhose notation for integrals is drawn directly from the work of Leibniz. While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour.