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Stewart Calculus Section 7.8

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- 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 ( )

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