Calculus II - 8
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Stewart Calculus Section 7.8

Stewart Calculus Section 7.8

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Calculus II - 8 Presentation Transcript

  • 1. 7.8 Improper IntegralsA standard definite integral: ( ) a b
  • 2. Improper integral of Type I: If ( ) exists for every , then ( ) = ( )provided this limit exists as a finite number.We call this improper integral convergent, otherwise divergent. a ∞
  • 3. Ex: and
  • 4. Ex: andSince = =
  • 5. Ex: andSince = = so = therefore =
  • 6. Ex: andSince = = so = therefore =However = =
  • 7. Ex: andSince = = so = therefore =However = = Since = , is divergent.
  • 8. Ex: andSince = = so = therefore =However = = Since = , is divergent. is convergent if > , otherwise divergent.
  • 9. Improper integral of Type I: If ( ) exists for every , then ( ) = ( ) If ( ) exists for every , then ( ) = ( ) If both ( ) and ( ) are convergent, ( ) = ( ) + ( )
  • 10. Ex: +
  • 11. Ex: + = + + + +
  • 12. Ex: + = + + + + Since = = +
  • 13. Ex: + = + + + + Since = = + = = +
  • 14. Ex: + = + + + + Since = = + = = + We have = + = +
  • 15. Improper integral of Type II: If ( ) is continuous on [ , ), then ( ) = ( ) if it exists as a finite number. a b
  • 16. Improper integral of Type II: If ( ) is continuous on [ , ), then ( ) = ( ) If ( ) is continuous on ( , ], then ( ) = ( ) + If both ( ) and ( ) are convergent, ( ) = ( ) + ( )
  • 17. Ex: and
  • 18. Ex: andSince = = + +
  • 19. Ex: andSince = = + + so =
  • 20. Ex: andSince = = + + so =However = ( )= + +
  • 21. Ex: andSince = = + + so =However = ( )= + + so is divergent.
  • 22. Comparison theorem: Suppose ( ) and ( ) are continuous, ( ) ( ) forIf ( ) is convergent, so is ( )If ( ) is divergent, so is ( )