1. 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals
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2. Copyright © Cengage Learning. All rights reserved.
8.8
Improper Integrals
2015
Copyright © Cengage Learning. All rights reserved.

3. Warm-Up
Evaluate the limit:

4. Objectives
Evaluate an improper integral that has an infinite limit of integration.
Evaluate an improper integral that has an infinite discontinuity.

5. Improper Integrals The definition of a definite integral
requires that the interval [a, b] be finite, and that f be continuous on [a, b].
This section examines a procedure for evaluating integrals that do not satisfy these requirements—usually because either one or both of the limits of integration are infinite,
or f has a finite number of infinite discontinuities in the interval [a, b].
Integrals that possess either property are said to be improper integrals.

6. Improper Integrals with Infinite Limits of Integration

7. Consider the integral
This can be interpreted as the area
of the shaded region in the figure:
Now, taking the limit as
The integral converges to 1.

8. Example 1 – An Improper Integral
Evaluate
Solution:

9. Example 1 – Solution
cont’d

10. Example 2 – Improper Integrals
Evaluate each improper integral:
Solution:

11. Example 3 – Using L’Hopital’s with an Improper Integral
Evaluate:
Now apply the definition of an improper integral:
Using L’Hopital’s on the limit:

13. Example 4 – Infinite upper and lower limits of integration
Evaluate: Convenient breaking point?
Solution:

15. You try
Evaluate:

16. Improper Integrals with Infinite Limits of Integration

17. Homework
Day 1: Section 8.8 pg #1-31 odd

18. Improper Integrals with Infinite Discontinuities
Day 2
Improper Integrals with Infinite Discontinuities

19. Try this:
Find the area of:
Graph?
To find the area under a curve that is discontinuous on the interval, divide the region at the discontinuity and compute the areas of the sub-regions.
The area is infinite between -1 and 1.

20. Improper Integrals with Infinite Discontinuities
A function f is said to have an infinite discontinuity at c if, from the right or left,

21. Improper Integrals with Infinite Discontinuities

22. Example 6 – An Improper Integral with an Infinite Discontinuity
Evaluate
Solution:
The integrand has an infinite discontinuity
at x = 0, as shown in Figure 8.24.
You can evaluate this integral as shown
below.
Figure 8.24

23. Example 7 – An Improper Integral
Evaluate
Solution:

24. If you don’t check to see that the integral is improper,
Example 8 – Try this one
Evaluate
Solution: The integral is improper because it has an infinite discontinuity at an interior point.
Rewrite as
If you don’t check to see that the integral is improper,
you’ll arrive at 3/8, which is incorrect.

26. Example:
(P is a constant.)
What happens here?
If then gets bigger and bigger as , therefore the integral diverges.
If then b has a negative exponent and ,
therefore the integral converges.

27. Improper Integrals with Infinite Discontinuities

28. Example 11 – An Application Involving A Solid of Revolution
The solid formed by revolving (about the x-axis) the unbounded region lying between the graph of f(x) = 1/x and the x-axis (x ≥ 1) is called Gabriel’s Horn. (See Figure 8.28.) Show that this solid has a finite volume and an infinite surface area.
Figure 8.28

29. Example 11 – Solution
Using the disk method and Theorem 8.5, you can determine the volume to be
The surface area is given by

30. Example 11 – Solution Because
cont’d
Because
on the interval [1, ), and the improper integral
diverges, you can conclude that the improper integral
also diverges.
So, the surface area is infinite.

31. Homework
Day 1: Section 8.8 pg. 585 #1-31 odd Day 2: Section 8.8 pg. 585 #33-49odd, 50, 53, 55, 67. Day 3: MMM 121,122,124