Calculus

:For other uses of the term calculus see calculus (disambiguation)

Fundamental theorem of calculus

The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, antiderivatives can be calculated with definite integrals, and vice versa.

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This connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter.

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This realization, made by both Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known.

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The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

Related Topics:
Antiderivative - Differential equation

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1st Fundamental Theorem of Calculus: If a function f is continuous on the interval and F is an antiderivative of f on the interval , then

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:int_{a}^{b} f(x),dx = F(b) - F(a).

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2nd Fundamental Theorem of Calculus: If f is continuous on an open interval I containing a, then, for every x in the interval,

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:rac{d}{dx}int_a^x f(t), dt = f(x).

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