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In 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 in calculus, with its inverse, differentiation, being the other. Given a function of a real variable and an interval of the real line, the definite integral : is defined informally as the signed area of the region in the -plane that is bounded by the graph of , the -axis and the vertical lines and . The area above the -axis adds to the total and that below the -axis subtracts from the total. Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term ''integral'' may also refer to the related notion of the antiderivative, a function whose derivative is the given function . In this case, it is called an ''indefinite integral'' and is written: : 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: if is a continuous real-valued function defined on a closed interval , then, once an antiderivative of is known, the definite integral of over that interval is given by : The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century, who 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. ==History== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「integral」の詳細全文を読む スポンサード リンク
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