Copyright © Cengage Learning. All rights reserved. 14 Further Integration Techniques and Applications of the Integral

Copyright © Cengage Learning. All rights reserved Improper Integrals and Applications

33 All the definite integrals we have seen so far have had the form f (x) dx, with a and b finite and f (x) piecewise continuous on the closed interval [a, b]. If we relax one or both of these requirements somewhat, we obtain what are called improper integrals. There are various types of improper integrals.

44 Integrals in Which a Limit of Integration is Infinite

55 Integrals in which one or more limits of integration are infinite can be written as Let’s concentrate for a moment on the first form,. What does the mean here? As it often does, it means that we are to take a limit as something gets large. Specifically, it means the limit as the upper bound of integration gets large.

66 Integrals in Which a Limit of Integration is Infinite Improper Integral with an Infinite Limit of Integration We define provided the limit exists. If the limit exists, we say that converges. Otherwise, we say that diverges.

77 Integrals in Which a Limit of Integration is Infinite Similarly, we define provided the limit exists. Finally, we define for some convenient a, provided both integrals on the right converge.

88 Integrals in Which a Limit of Integration is Infinite Quick Example Converges

99 Example 1 – Future Sales of CDs In 2006, music downloads were starting to make inroads into the sales of CDs. Approximately 140 million CD albums were sold in the first quarter of 2006, and sales declined by about 3.5% per quarter for the following 2 years. Suppose that this rate of decrease were to continue indefinitely. How many CD albums, total, would be sold from the first quarter of 2006 on?

10 Example 1 – Solution Recall that the total sales between two dates can be computed as the definite integral of the rate of sales. So, if we wanted the sales between the first quarter of 2006 and a time far in the future, we would compute with a large M, where s(t) is the quarterly sales t quarters after the first quarter of Because we want to know the total number of CD albums sold from the first quarter of 2006 on, we let M → that is, we compute

11 Example 1 – Solution Because sales of CD albums are decreasing by 3.5% per quarter, we can model s(t) by s(t) = 140(0.965) t million CD albums where t is the number of quarters since the first quarter of Total sales from the first quarter of 2006 on cont’d

12 Example 1 – Solution ≈ million CD albums. cont’d

13 Integrals in Which the Integrand Becomes Infinite

14 Example 2 – Integrand Infinite at One Endpoint Calculate Solution: Notice that the integrand approaches as x approaches 0 from the right and is not defined at 0. This makes the integral an improper integral. Figure 22 shows the region whose area we are trying to calculate; it extends infinitely vertically rather than horizontally. Figure 22

15 Example 2 – Solution Now, if 0 < r < 1, the integral is a proper integral because we avoid the bad behavior at 0. This integral gives the area shown in Figure 23. If we let r approach 0 from the right, the area in Figure 23 will approach the area in Figure 22. cont’d Figure 23

16 Example 2 – Solution So, we calculate = 2. Thus, we again have an infinitely long region with finite area. cont’d

17 Integrals in Which the Integrand Becomes Infinite Improper Integral in Which the Integrand Becomes Infinite If f (x) is defined for all x with a < x  b but approaches as x approaches a, we define provided the limit exists.

18 Integrals in Which the Integrand Becomes Infinite Similarly, if f (x) is defined for all x with a ≤ x < b but approaches as x approaches b, we define provided the limit exists. In either case, if the limit exists, we say that converges. Otherwise, we say that diverges.