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# Calculus II - 8

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Stewart Calculus Section 7.8

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• ### Calculus II - 8

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