7.8 Improper IntegralsA standard definite integral:                    ( )    a                           b
Improper integral of Type I: If       ( )   exists for every         , then                ( )   =            ( )provided ...
Ex:   and
Ex:     andSince   =     =
Ex:     andSince   =         = so           =       therefore   =
Ex:       andSince     =           = so               =       therefore   =However       =           =
Ex:       andSince     =               = so               =           therefore             =However       =              ...
Ex:             andSince            =               = so                      =           therefore              =However ...
Improper integral of Type I: If    ( )       exists for every             , then                 ( )    =              ( )...
Ex:      +
Ex:          +              =       +      +           +       +
Ex:             +                 =               +         +               +           + Since                       =   ...
Ex:             +                 =               +         +               +           + Since                       =   ...
Ex:             +                 =                   +         +               +                   + Since               ...
Improper integral of Type II: If   ( )   is continuous on   [ , ),   then               ( )   =            ( ) if it exist...
Improper integral of Type II: If   ( )   is continuous on        [ , ),    then                  ( )     =             ( )...
Ex:   and
Ex:         andSince             =       =        +             +
Ex:         andSince             =       =        +             + so         =
Ex:           andSince               =                   =          +                 + so           =However             ...
Ex:            andSince                 =                   =          +                   + so            =However       ...
Comparison theorem: Suppose     ( )   and    ( )   are continuous,     ( )     ( )         forIf         ( )     is conver...
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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 ( )

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