Eroxl's Notes
The Fundamental Theorem of Calculus

The fundamental theorem of calculus relates derivatives and integrals together. It describes how derivatives and integrals are inverses of each other. The fundamental theorem of calculus is composed of two parts the first which describes antiderivatives and the second which describes integrals.

First Part

If is continuous on the closed interval then

Examples

Find if

Find

Note

Notice how the chain rule was used to make it possible to take the derivative with respect to instead of .

Find if

Second Part

If is a function defined on the closed interval and is a continuous functionon which is an antiderivative of over the open interval then

Examples

Evaluate

Evaluate

Evaluate