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• Integral abstract "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 [a, b] of the real line, the definite integral is defined informally to be the signed 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 related 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:However, 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 f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given byIntegrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The founders of 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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• Integral align "center".
• Integral align "right".
• Integral alt "Lower Darboux sum example".
• Integral alt "Riemann integral approximation example".
• Integral alt "Riemann sum convergence".
• Integral alt "Upper Darboux sum example".
• Integral caption "Darboux lower sums of the function y = x2".
• Integral caption "Darboux upper sums of the function y = x2".
• Integral caption "Integral example with irregular partitions".
• Integral caption "Riemann sums converging".
• Integral direction "horizontal".
• Integral direction "vertical".
• Integral hasPhotoCollection Integral.
• Integral header "Darboux sums".
• Integral headerAlign "center".
• Integral id "p/i051340".
• Integral image "Integral Riemann sum.png".
• Integral image "Riemann Integration and Darboux Lower Sums.gif".
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• Integral image "Riemann sum convergence.png".
• Integral title "Integral".
• Integral width "200".
• Integral width "300".
• Integral subject Category:Functions_and_mappings.
• Integral subject Category:Integrals.
• Integral subject Category:Linear_operators_in_calculus.
• Integral comment "Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus.".
• Integral label "Całka".
• Integral label "Integración".
• Integral label "Integral".
• Integral label "Integral".
• Integral label "Integrale".
• Integral label "Intégration (mathématiques)".
• Integral label "Интеграл".
• Integral label "تكامل".
• Integral label "积分".
• Integral label "積分法".
• Integral sameAs Integrál.
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• Integral sameAs Integral.
• Integral sameAs Intégration_(mathématiques).
• Integral sameAs Integral.
• Integral sameAs Integrale.
• Integral sameAs 積分法.
• Integral sameAs 적분.
• Integral sameAs Całka.
• Integral sameAs Integral.
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• Integral depiction Integral_example.svg.
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