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See also: Proof of the Fundamental Theorem 
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Fundamental Theorem of Calculus

FUNDAMENTAL THEOREM OF CALCULUS

Suppose f is continuous on its domain, which is an open interval I.

(i)

For each point a in I, the definite integral of f from a to x considered as a function of x is an antiderivative of f. That is,

04_integration-128.gif
(ii)

If F is any antiderivative of f, then for any two points (a, b) in I the definite integral of f from a to b is equal to the difference F(b) - F(a),

04_integration-129.gif

The Fundamental Theorem of Calculus is important for two reasons. First, it shows the relation between the two main notions of calculus: the derivative, which corresponds to velocity, and the integral, which corresponds to area. It shows that differentiation and integration are "inverse" processes. Second, it gives a simple method for computing many definite integrals.

Example 1: Antiderivative of a Line

Example 2: Antiderivative of a Parabola

Example 3: Antiderivative of Velocity

Example 4.

Example 5.

Example 6.

Example 7.

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Last Update: 2006-11-25