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Further Integration Techniques and Applications of the Integral
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7.5
Improper Integrals and Applications
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3. Improper Integrals and Applications
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.

4. Integrals in Which a Limit of Integration is Infinite

5. Integrals in Which a Limit of Integration is Infinite
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.

6. 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.

7. 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.

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

9. Example 1 – Future Sales of CDs
In 2009, music downloads were starting to make inroads into the sales of CDs. Approximately 290 million CD albums were sold in 2009, and sales declined by about 23% per year the following year. Suppose that this rate of decrease were to continue indefinitely. How many CD albums, total, would be sold from the first quarter of 2009 on?

10. Example 1 – Solution
We know 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 2009 and a time far in the future, we would compute with a large M, where s(t) is the annual sales t quarters after the first quarter of 2009.
Because we want to know the total number of CD albums sold from 2009 on, we let M → that is, we compute

11. Example 1 – Solution
cont’d
Because sales of CD albums are decreasing by 23% per year, we can model s(t) by
s(t) = 290(0.77)t million CD albums
where t is the number years since 2009.
Total sales from 2009 on

12. Example 1 – Solution
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 23 shows the region
whose area we are trying to calculate; it extends infinitely
vertically rather than horizontally.
Figure 23

15. Example 2 – Solution
cont’d
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 24.
If we let r approach 0 from
the right, the area in
Figure 24 will approach
the area in Figure 23.
Figure 24

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

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.