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 of the real line, the definite integral
is defined informally to be the area of the region in the 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 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:
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, then, once an antiderivative F of f is known, the definite integral of f 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 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.
Other articles related to "integral":
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... In Calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the form then for the ... This important result is particularly useful in the differentiation of integral transforms ... Whether Leibniz's Integral Rule applies is essentially a question about the interchange of limits ...
... Enlightenment shares important principles of, and is often associated with, Integral Theory, in that both combine spiritual and scientific insights to create a comprehensive understanding of humanity and the universe ... Evolutionary Enlightenment and Integral Theory share an understanding that ultimate reality consists of a non-dual union of emptiness and form—with form being subject to ... One outcome of this insight, which Integral thinkers have codified and mapped, is that there are hierarchical stages of development along a deep-time evolutionary continuum, and that the interdependence between ...
... In chaos theory, the correlation integral is the mean probability that the states at two different times are close where is the number of considered states, is a threshold distance, a norm (e ... The correlation integral is used to estimate the correlation dimension ... An estimator of the correlation integral is the correlation sum ...
... First we show that if is an antiderivative of on, then it has the path integral property given above ... Given any piecewise C1 path, one can express the path integral of over as By the chain rule and the fundamental theorem of calculus one then has Therefore the integral of over does not depend on the ...
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