2. 8.4 Improper Integrals

3. Quick Review Evaluate the integral.

4. Quick Review Find the domain of the function.

5. What you’ll learn about Infinite Limits of Integration Integrands with Infinite Discontinuities Test for Convergence and Divergence Essential Question What techniques can be used to extend integration techniques to cases where the interval of integration [a,b] is not finite or where integrands are not continuous.

6. Improper Integrals with Infinite Integration Limits Integrals with infinite limits of integration are improper integrals. 1. If f (x) is continuous on [a, ∞), then 2. If f (x) is continuous on (– ∞, b], then 3. If f (x) is continuous on (– ∞, ∞), then

7. Example Evaluating an Improper Integral on [1,∞) 1.Does the following improper integral converge or diverge? Thus, the integral diverges.

8. Example Using L’Hôpital’s Rule with Improper Integrals Use integration by part to evaluate the definite integral.

9. Example Evaluating an Integral on (-∞,∞) Evaluate each improper integral:

10. Improper Integrals with Infinite Discontinuities Integrals of functions that become infinite at a point within the interval of integration are improper integrals. 1. If f (x) is continuous on (a, b], then 2. If f (x) is continuous on [ a, b), then 3. If f (x) is continuous on [a, c) U (c, b ],then

11. Example Infinite Discontinuity at an Interior Point The integrand has a vertical asymptote at x = 1 and is continuous on [0, 1) and (1, 2]. Evaluate each improper integral:

12. Comparison Test Let f and g be continuous on [a, ∞) with 0 a. Then

13. Example Finding the Volume of an Infinite Solid 5.Find the volume of the solid obtained by revolving the following curve about the x-axis. Using tabular integration we get:

14. Example Finding the Volume of an Infinite Solid 5.Find the volume of the solid obtained by revolving the following curve about the x-axis.

15. Pg. 467, 8.4 #1-43 odd