In this chapter, we use the problem of calculating the area of a region to introduce the concept of the definite integral. Although in the beginning, the definite integral seems very different from the indefinite integral, introduced in the previous chapter, the Fundamental Theorem of Calculus reveals a deep connection between them. In this chapter, we apply the definite integral to find the net change of a variable and study the rectilinear motion, and to find the average value of a function. Some other applications of the definite integral will be discussed in the next chapter.

In this chapter, we will talk about the following topics:

8.1 Definition of the Definite Integral

8.2 Properties of Definite Integrals

8.3 The Fundamental Theorem of Calculus

8.4 Derivatives of Integrals

8.5 Evaluation of Definite Integrals by Substitution