## Calculus AB and Calculus BC

## CHAPTER 6 Definite Integrals

**Concepts and Skills**

In this chapter, we will review what definite integrals mean and how to evaluate them. We’ll look at

• the all-important Fundamental Theorem of Calculus;

• other important properties of definite integrals, including the Mean Value Theorem for Integrals;

• analytic methods for evaluating definite integrals;

• evaluating definite integrals using tables and graphs;

• Riemann sums;

• numerical methods for approximating definite integrals, including left and right rectangular sums, the midpoint rule, and the trapezoid rule;

• and the average value of a function.

For BC students, we’ll also review how to work with integrals based on parametrically defined functions.

### A. FUNDAMENTAL THEOREM OF CALCULUS (FTC); DEFINITION OF DEFINITE INTEGRAL

If *f* is continuous on the closed interval [*a*, *b*] and *F* *′* = *f*, then, according to the Fundamental Theorem of Calculus,

**Definite integrals**

Here is the *definite integral of f from a to b; f* (*x*) is called the *integrand;* and *a* and *b* are called respectively the *lower* and *upper limits of integration.*

This important theorem says that if *f* is the derivative of *F* then the definite integral of *f* gives the net change in *F* as *x* varies from *a* to *b.* It also says that we can evaluate any definite integral for which we can find an antiderivative of a continuous function.

By extension, a definite integral can be evaluated for any function that is bounded and piecewise continuous. Such functions are said to be *integrable.*