Chapter 17
Area Under a Curve¹

In Chapters 14 and 15 we learned how to find the rate of change function, the derivative, of various functions. We used the derivative in business applications to find such things as marginal cost or marginal profit functions and to study optimization of cost or revenue. In this chapter we do the opposite: For example, instead of beginning with the cost function c and then finding the marginal cost function by applying the rules of differentiation to c, we begin with and find c by reversing the rules of differentiation. This process is called antidifferentiation or finding the indefinite integral. Mathematics and science majors usually take an entire course in differentiation and then follow it with another course devoted to integration. In these courses, students study a rather amazing idea, called the Fundamental Theorem of Calculus; namely, an anitderivative c of a derivative function (found by reversing the rules of differentiation) is intimately connected to the area under the graph of the derivative function f′(x). Although we will study some of the basic rules of antidifferentiation in order to find the area under various curves by using the Fundamental Theorem of Calculus, we will also use spreadsheets to find approximate numerical answers to finding the area, approximations that serve us quite well in real-life situations. This process is called numerical integration. Indeed, for some important functions there is no known way of finding their antiderivative and, as a result, numerical integration is the only way we have of finding the area under the graph of these particular functions.

1. In the first section of the chapter, we will use both numerical integration and the Fundamental Theorem of Calculus to find the area under a curve.
2. In the second section, we apply finding the area under a curve to some business applications.