Barrons AP Calculus - David Bock

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.

B. PROPERTIES OF DEFINITE INTEGRALS