HSC Higher Mathematics 1st Paper Note 10th Chapter Integration. This article is about the concept of definite integrals in calculus. For the indefinite integral, see antiderivative. For the set of numbers, see integer. For other uses, see Integral (disambiguation). 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 of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral:
HSC Higher Mathematics 1st Paper Note 10th Chapter Integration
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. Bernhard Riemann gave a rigorous mathematical definition of integrals. It 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 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 generalized. 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.
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