## 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:

What is the probability that a random person will score between 1200 and 1250?

What is

What is

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 4.4 The Fundamental Theorem of Calculus (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

#### Interesting Application

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel