In this chapter we describe important applications of integration, and we develop procedures for evaluating both definite and indefinite integrals. We begin with the physical problem of finding the center of mass of an irregular object. This problem illustrates the important of the divide-and-conquer process that leads to the definite integral.

Next we discuss the process of evaluating definite integrals — first numerically and then symbolically. Our numerical methods start with the approximating-sum definition and lead to the type of method likely to be built into your calculator or computer software. A fundamental technique for symbolic integration is substitution for the variable of integration, which is closely related to the Chain Rule for differentiation. Later we consider the implications of the differentiation Product Rule, which leads to a sophisticated technique called "integration by parts."

With the necessary tools in place, we apply integration to the problem of representing a complicated periodic function as a sum of sine and cosine functions. This process, called "harmonic analysis," is basic in such diverse areas as synthesis of music and analysis of heat flow.