4.4 The Fundamental Theorem of Calculus

Summary

The Fundamental Theorem of Calculus relates derivatives and definite integrals. It also gives a practical way to evaluate many definite integrals without resorting to the limit definition. Hence, it is truly fundamental in the study of calculus. This section is devoted to a statement and proof of the Fundamantal Theorem, and some examples of its use.

By the end of your studying, you should know:

A statement of the Fundamental Theorem Part I—antiderivative.
A statement of the Fundamental Theorem Part II—evaluation.
How to use the Fundamental Theorem to evaluate definite integrals.

On-screen applet instructions: The applet computes the difference quotient of F for a given x. Use the slider to control the value of h. You can choose different values of x from the pull-down menu.

Examples

The Scholastic Aptitude Test (SAT) is rescaled so that the scores of n people, ranging from 0 to 1600, fit a distribution in the shape of the following function: