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Ms. Battaglia AP Calculus

8-8 Improper Integrals Objective: Evaluate an improper integral that has an infinite limit of integration and an infinite discontinuity. Ms. Battaglia AP Calculus. Improper Integrals (Just look at the way that integral is holding its fork!).

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Ms. Battaglia AP Calculus

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  1. 8-8 Improper IntegralsObjective: Evaluate an improper integral that has an infinite limit of integration and an infinite discontinuity. Ms. Battaglia AP Calculus

  2. Improper Integrals(Just look at the way that integral is holding its fork!) Definite integrals are improper when the go infinitely far up, down, right or left. Ex: (one or more vertical asymptotes) Ex: one or both of the limits of integration is infinite

  3. Def of Improper Integrals with Infinite Integration Limits • If f is continuous on the interval [a,∞), then • If f is continuous on the interval (-∞,b], then • If f is continuous on the interval (-∞,∞), then where c is any real number. In the 1st two cases, the improper integral converges if the limit exists- otherwise, it diverges. Third case: left diverges if either of the right diverge.

  4. Def of Improper Integrals with Infinite Discontinuities • If f is continuous on the interval [a,b), and has an infinite discontinuity at b, then • If f is continuous on the interval (a,b], and has an infinite discontinuity at a, then • If f is continuous on the interval [a,b], except for some c in (a,b) at which f has an infinite discontinuity, then where c is any real number. In the 1st two cases, the improper integral converges if the limit exists- otherwise, it diverges. Third case: left diverges if either of the right diverge.

  5. An Improper Integral that Diverges Evaluate

  6. Improper Integrals That Converge Evaluate each improper integral. a. b.

  7. Example What is the area under from 0 to 1?

  8. Example What is the area under from 0 to 1?

  9. Example Evaluate

  10. Example Evaluate

  11. Example Evaluate

  12. Classwork/Homework • Read 8.8 Page 587 #5, 7, 9-14, 19, 22, 31

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