Fundamental Theorems of Calculus (FTC)

The theorem that establishes the connection between the two branches of calculus: differential calculus and integral calculus. The fundamental theorem of calculus is typically given in two parts. It says the following:

Suppose  f  is continuous on [a, b]. Then:

(1) The function

is an antiderivative of f, i.e., g'(x) = f(x).

(2) (Evaluation Theorem) If F is an antiderivative of f, i.e. F'(x) = f(x), then

 

The two parts of the theorem can be rewritten like this:

(1) 

(2) 

Important:  The theorem states that integration and differentiation are inverse operation. For the derivative of an integral of a function yields the original function, and the integral of a derivative also yields the function originally differentiated (up to a constant).

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credit: James Stewart©2013 www.FroydWess.com

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