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**Integration**is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function

*f*of a real variable

*x*and an interval [

*a*,

*b*] of the real line, the

**definite integral**

*xy*-plane bounded by the graph of

*f*, the

*x*-axis, and the vertical lines

*x*=

*a*and

*x*=

*b*, such that area above the

*x*-axis adds to the total, and that below the

*x*-axis subtracts from the total.

The term

*integral*may also refer to the related notion of the antiderivative, a function

*F*whose derivative is the given function

*f*. In this case, it is called an

*indefinite integral*and is written:

*definite integrals*.

The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th 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

*a*,

*b*] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

see this diagram to understand Integration .

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