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Integral calculus)
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
is an important concept in
mathematics and, together with its inverse,
differentiation, is one of the two main operations in
calculus. Given a
function ƒ of a
real variable x and an
interval [
a,
b] of the
real line, the
definite integral
is defined informally to be the net 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
ƒ. In this case, it is called an
indefinite integral and is written:
The integrals discussed in this article are termed
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
ƒ is a continuous real-valued function defined on a
closed interval [
a,
b], then, once an antiderivative
F of
ƒ is known, the definite integral of
ƒ over that interval is given by
Integrals and derivatives became the basic tools of calculus, with numerous applications in science and
engineering. The founders of the calculus thought of the integral as an infinite sum of rectangles of
infinitesimal width. A rigorous mathematical definition of the integral was given by
Bernhard Riemann. It is based on a limiting procedure which approximates the area of a
curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A
line integral is defined for functions of two or three variables, and the interval of integration [
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.
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