Integral

: This article deals with the concept of an integral in calculus. For other meanings of "integral" see integration and integral (disambiguation).

Related Topics:
Calculus - Integration - Integral (disambiguation)

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In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. The process of finding integrals is integration, in its mathematical meaning. Unlike the closely-related process of differentiation, there are several possible definitions of integration, with different technical underpinnings. They are, however, compatible. Any two different ways of integrating a function will give the same result if they are both defined.

Related Topics:
Calculus - Function - Area - Mass - Volume - Sum - Total - Differentiation

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Intuitively, the integral of a continuous, positive real-valued function f of one real variable x between a left endpoint a and a right endpoint b represents the area bounded by the lines x=a, x=b, the x-axis, and the curve defined by the graph of f. More formally, if we let

Related Topics:
Continuous - Positive

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: S= {(x,y) in mathbb{R}^2:a leq x leq b ,0 leq y leq f(x)},

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then the integral of f between a and b is the measure of S.

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Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written int_a^b f(x),dx. The ∫ sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating, and dx is a notation for the variable of integration. Historically, dx represented an infinitesimal quantity, and the long s stood for "sum". However, modern theories of integration are built from different foundations, and the traditional symbols have become no more than notation.

Related Topics:
Leibniz - Long s - Interval - Infinitesimal - Notation

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As an example, if f is the constant function f(x)=3, then the integral of f between 0 and 10 is the area of the rectangle bounded by the lines x=0, x=10, y=0, and y=3. The area is the width of the rectangle times its height, so the value of the integral is 30.

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Integrals can be taken over regions other than intervals. In general, the integral over a set E of a function f is written ∫Ef(x)dx. Here x need not be a real number, but, for instance, a vector in R3. Fubini's theorem shows that such integrals can be rewritten as an iterated integral. In other words, the integral can be calculated by integrating one coordinate at a time.

Related Topics:
Set - Vector - Fubini's theorem

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If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. Integrals are sometimes called definite integrals to emphasize that they result in a number, not another function. This is to distinguish them from indefinite integrals, which are another name for an antiderivative. If the domain of the function to be integrated is the real numbers, and if the region of integration is an interval, then the greatest lower bound of the interval is called the lower limit of integration, and the least upper bound is called the upper limit of integration.

Related Topics:
Antiderivative - Real number - Interval - Greatest lower bound - Least upper bound

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