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General Information On The Function And Importance Video

In mathematicsan 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, differentiationis the other.

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It is denoted. 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 antiderivativecalled an indefinite integrala function F whose derivative is the given function f. In this case, it Infkrmation written:.

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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 f is a continuous real-valued function defined on a closed interval [ ab ]then once an antiderivative F of f is known, the definite integral of f over that interval is given by. The principles of integration were formulated independently by Isaac Newton please click for source 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 Importancf a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical Impkrtance.

Beginning General Information On The Function And Importance 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 integration [ ab ] is replaced by a curve connecting the two endpoints.

In a surface integralthe curve is replaced by a piece of a surface in three-dimensional space. The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus ca. A similar method was independently developed in China around the 3rd century AD by Liu Huiwho used it to find the area Genera, the circle.

This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. The next significant advances in integral calculus did not begin to appear until the 17th century.

Barrow provided the first proof of the fundamental theorem of calculus. The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton.

The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals.

In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculusit allowed for precise analysis of functions within continuous domains.]