# Integral

### From Wikipedia, the free encyclopedia

**Integration** is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function *ƒ* of a real variable *x* and an interval [*a*,*b*] of the real line, the **integral**

is defined informally to be the signed area of the region in the *xy*-plane bounded by the graph of *ƒ*, the *x*-axis, and the vertical lines *x* = *a* and *x* = *b*.

The term "integral" may also refer to the notion of antiderivative, a function *F* whose derivative is the given function *f*. In this case it is called an **indefinite integral**, while the integrals discussed in this article are termed **definite integrals**. Some authors maintain a distinction between antiderivatives and indefinite integrals.

The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late seventeenth century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if *f* is a continuous real-valued function defined on a closed interval [*a*, *b*], then, once an antiderivative *F* of *f* is known, the definite integral of *f* over that interval is given by