Improper Integrals

1. Improper Integrals Section 8.3a

2. Consider the infinite region in the first quadrant that lies under the given curve: We can now calculate the finite value of this area! First we find the area of the portion of the region that is bounded on the right by x = b: Then find the limit of this value as b approaches infinity…

3. Consider the infinite region in the first quadrant that lies under the given curve: We can now calculate the finite value of this area! Then find the limit of this value as b approaches infinity… So the area under the curve from zero to infinity is

4. Improper Integrals with Infinite Integration Limits Integrals with infinite limits of integration are improper integrals. 1. If is continuous on , then 2. If is continuous on , then 3. If is continuous on , then where c is any real number.

5. Improper Integrals with Infinite Integration Limits Integrals with infinite limits of integration are improper integrals. In parts 1 and 2, if the limit is finite the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of the equation converges if both improper integrals on the right-hand side converge, otherwise it diverges and has no value.

6. Practice Problems Does the improper integral converge or diverge? Apply our new rule: Because this limit does not have a finite value, the initial integral diverges.

7. Practice Problems Evaluate: Use part 3 of the new rule:

8. Practice Problems Evaluate: Use part 3 of the new rule: Thus,

9. Practice Problems For what values of p does the given integral converge? When the integral does converge, what is its value? We have already shown that for p = 1 the integral diverges… For other values of p:

10. Consider the infinite region in the first quadrant that lies under the given curve from x = 0 to x = 1: We can now calculate the finite value of this area! First we find the area of the portion from a to 1: Then find the limit of this value as a approaches zero from the right…

11. Consider the infinite region in the first quadrant that lies under the given curve from x = 0 to x = 1: We can now calculate the finite value of this area! Then find the limit of this value as a approaches zero from the right… So the area under the curve from zero to one is 2!!!

12. Improper Integrals with Infinite Discontinuities Integrals of functions that become infinite at a point within the interval of integration are improper integrals. 1. If is continuous on , then 2. If is continuous on , then 3. If is continuous on , then

13. Improper Integrals with Infinite Discontinuities Integrals of functions that become infinite at a point within the interval of integration are improper integrals. In parts 1 and 2, if the limit is finite the improper integral converges and the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. In part 3, the integral on the left-hand side of the equation converges if both improper integrals on the right-hand side converge, otherwise it diverges and has no value.

14. Practice Problems The integrand has a VA at x = 1. Use part 3 of the new rule: Evaluate:

15. Practice Problems The integrand has a VA at x = 1. Use part 3 of the new rule: Evaluate:

16. Practice Problems The integrand has a VA at x = 2. Use part 3 of the new rule: Evaluate:

17. Practice Problems The integrand has a VA at x = 2. Use part 3 of the new rule: Evaluate: Because these limits do not have finite values, we can conclude that the original integral diverges and has no value.