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Understanding Improper Integrals: Concepts and Examples

Learn about improper integrals with infinite limits, determine convergence or divergence, and evaluate examples with step-by-step solutions.

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Understanding Improper Integrals: Concepts and Examples

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  1. 9-4 Improper Integrals

  2. b So, the area under the curve from 0 to  is

  3. Def: Integrals with infinite limits of integration are improper integrals. • If f (x) is continuous on [a, ), then • If f (x) is continuous on (–, b], then • If f (x) is continuous on (–, ), then where c is any real #

  4. *For parts (1) and (2), if the limit is a finite value then it converges; if no limit exists then it diverges *For part (3), both integrals on the right must converge for it to converge, otherwise it diverges Ex 1) Express the improper integral in terms of limits of definite integrals and then evaluate the integral. choose c = 0 Diverges

  5. Ex 2) Does the improper integral converge or diverge? Diverges

  6. Ex 3) Evaluate or state that it diverges. Partial fractions let x = –3 let x = –1 –2B = 2 2A = 2 B = –1 A = 1

  7. Ex 3) cont. 0 So, 0

  8. Ex 4) Evaluate or state that it diverges. (+) Use tabular! (–) So,

  9. Ex 5) Evaluate So,

  10. homework Pg. 461 # 4, 8, 10, 16 Pg. 471 # 1, 3, 5, 7, 11, 13, 17, 19, 21

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