MEDIA - Integral Compatible Computer Devices

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the concept of definite integrals in calculus|the indefinite integral|antiderivative|the set of numbers|integer|other uses|Integral (disambiguation)thumb|300px|A definite integral of a function can be represented as the signed area of the region bounded by its graph.|alt=Definite integral exampleIn mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus; its inverse operation, differentiation, is the other. Given a functionof a real variableand an intervalof the real line, the definite integral offromtocan be interpreted informally as the signed area of the region in theplane that is bounded by the graph of theaxis and the vertical linesi> a andi> b . It is denoted : $\backslash int\_a^b\; f(x)\backslash ,dx.$ The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, called an indefinite integral , a functionwhose derivative is the given function In this case, it is written: :$F(x)\; =\; \backslash int\; f(x)\backslash ,dx.$ The integrals discussed in this article are those termed definite integrals. It is the fundamental theorem of calculus that connects differentiation with the definite integral: ifis a continuous real-valued function defined on a closed interval then once an antiderivativeofis known, the definite integral ofover that interval is given by :$\backslash int\_a^b\; \backslash ,\; f(x)\; dx\; =\; \backslash left[\; F(x)\; \backslash right]\_a^b\; =\; F(b)\; -\; F(a)\; \backslash ,\; .$ The principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous mathematical definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the 19th 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 generalized. A line integral is defined for functions of two or more variables, and the interval of integrationis replaced by a curve connecting the two endpoints. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.