Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,^[a] the other being 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.

This article is about the concept of definite integrals in calculus. For the indefinite integral, see antiderivative. For the set of numbers, see integer. For other uses, see Integral (disambiguation).

A definite integral computes 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 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 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 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 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 over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the 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 in three-dimensional space.

The , which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the Riemann integral. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.

Darboux integral

The , an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.

Riemann–Stieltjes integral

The , further developed by Johann Radon, which generalizes both the Riemann–Stieltjes and Lebesgue integrals.

Lebesgue–Stieltjes integral

The , which subsumes the Lebesgue integral and Lebesgue–Stieltjes integral without depending on measures.

Daniell integral

The , used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933.

Haar integral

The , variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.

Henstock–Kurzweil integral

The and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion.

Itô integral

The , which is a kind of Riemann–Stieltjes integral with respect to certain functions of unbounded variation.

Young integral

The integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both semimartingales and processes such as the fractional Brownian motion.

rough path

The , a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.

Choquet integral

The , an extension of the Lebesgue integral to a more general class of functions, namely, those with a domain that is a Banach space.

Bochner integral

Properties[edit]

Linearity[edit]

The collection of Riemann-integrable functions on a closed interval $[a, b]$ forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration

Computation[edit]

Analytical[edit]

The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. Let $f (x)$ be the function of $x$ to be integrated over a given interval $[a, b]$ . Then, find an antiderivative of $f$ ; that is, a function $F$ such that $F' = f$ on the interval. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus,

Examples[edit]

Using the fundamental theorem of calculus[edit]

The fundamental theorem of calculus allows straightforward calculations of basic functions:

– Equations with an unknown function under an integral sign

Integral equation

– Mathematical symbol used to denote integrals and antiderivatives

Integral symbol

Lists of integrals

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Integral"

Wolfram Alpha.