Stewart Calculus Section 7.8

1. 7.8 Improper Integrals
A 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: and
Since = =

5. Ex: and
Since = =
so = therefore =

6. Ex: and
Since = =
so = therefore =
However = =

7. Ex: and
Since = =
so = therefore =
However = =
Since = , is divergent.

8. Ex: and
Since = =
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: and
Since = =
+ +

19. Ex: and
Since = =
+ +
so =

20. Ex: and
Since = =
+ +
so =
However = ( )=
+ +

21. Ex: and
Since = =
+ +
so =
However = ( )=
+ +
so is divergent.

22. Comparison theorem:
Suppose ( ) and ( ) are continuous,
( ) ( ) for
If ( ) is convergent, so is ( )
If ( ) is divergent, so is ( )