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,

Image

Definite integrals

Here Image 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.