Section 8.8 – Improper Integrals

1. Section 8.8 – Improper Integrals

2. The Fundamental Theorem of Calculus If f is continuous on the interval [a,b] and F is any function that satisfies F '(x) = f(x) throughout this interval then REMEMBER: [a,b] is a closed interval.

3. Improper Integral Areas of unbounded regions also arise in applications and are represented by improper integrals. Consider the following integral: Intuitively it appears any unbounded region should have infinite area.

4. Numerical Investigation Complete the table for with the table below. Numerically, it appears 1250 000

5. Definition: The Integral Diverges If the limit fails to exist, the improper integral diverges. For instance:

6. Improper Integral Areas of unbounded regions also arise in applications and are represented by improper integrals. Consider the following integral: Can the unbounded region have finite area?

7. Numerical Investigation Complete the table for with the table below. 0.632 0.865 0.950 Numerically, it appears 0.993 1.000 1.000 1.000 1.000 1.000 1.000

8. Definition: The Integral Converges If the limit is finite, the improper integral converges and the limit is the value of the improper integral. For instance:

9. “Horizontal” Improper Integrals In a horizontal improper integral, the left limit of integration vanishes into or the right limit vanishes into , or both limits vanish in respective directions and we integrate over the whole x-axis. If is continuous on the entire interval, then Both integrals must converge for the sum to converge. Note: It can be shown the value of c above is unimportant. You can evaluate the integral with any choice.

10. Example 1 Analytically evaluate .

11. Improper Integral Technique The technique for evaluating an improper integral “properly” is to evaluate the integral on a bounded closed interval where the function is continuous and the Fundamental Theorem of Calculus applies, then take the offending end of the interval to the limit. On any free-response question, always use the limit notation to evaluate improper integrals. While the statement below may involve less writing, it is mathematically incorrect and will lose you points: DO NOT WRITE THIS!

12. Example 2 Analytically evaluate . Use L'Hôpital's Rule

13. Example 2 Analytically evaluate .

14. White Board Challenge Evaluate:

15. The Fundamental Theorem of Calculus If f is continuous on the interval [a,b] and F is any function that satisfies F '(x) = f(x) throughout this interval then REMEMBER: f must be a continuous function.

16. Improper Integral Areas of unbounded regions also arise in applications and are represented by improper integrals. Consider the following integral: Can the unbounded region have finite area?

17. Numerical Investigation Complete the table for with the table below. 0.586 0.735 0.905 Numerically, it appears 1.368 1.937 1.98 1.994

18. “Vertical” Improper Integrals In a vertical improper integral, we integrate over a closed interval but the function has a vertical asymptote at one or both ends of the interval. If is continuous on the entire interval except one or both endpoints, then Both integrals must converge for the sum to converge. Note: It can be shown the value of c above is unimportant. You can evaluate the integral with any choice.

19. Example 1 Analytically evaluate .

20. Example 2 Analytically evaluate . Since the integral is infinite, it diverges (does not exist).

21. Example 3 Analytically evaluate .

22. White Board Challenge Evaluate: