Stewart Calculus Section 7.7

1. 7.7 Approximate Integration
Some functions have no explicit anti-
derivative, e.g. − , and so on.
In practice we need to evaluate the integral
anyway, so numerical approximate methods
are needed.

2. Basic Idea: Use the summation of area in
subintervals to approximate the integral.

3. Basic Idea: Use the summation of area in
subintervals to approximate the integral.
Midpoint Rule
Trapezoidal Rule
Simpson’s Rule

4. Midpoint Rule:
( ) [ (¯ ) + (¯ ) + · · · + (¯ )]
= ¯ = ( + )

5. Trapezoidal Rule:
( ) [ ( )+ ( ) + ··· + ( )+ ( )]
=

6. Simpson’s Rule:
( ) [ ( )+ ( )+ ( )+ ( ) + ···
+ ( )+ ( )+ ( )]
= n is e
ven!

7. Midpoint Rule:
( ) [ (¯ ) + (¯ ) + · · · + (¯ )]
¯ = ( + )
Trapezoidal Rule:
( ) [ ( )+ ( ) + ··· + ( )+ ( )]
Simpson’s Rule:
( ) [ ( )+ ( )+ ( )+ ( ) + ···
+ ( )+ ( )+ ( )]

8. Error Bound of Midpoint Rule:
Suppose | ( )| for
( )
then | |
Error Bound of Trapezoidal Rule:
Suppose | ( )| for
( )
then | |
Error Bound of Simpson’s Rule:
Suppose | ( )| for
( )
then | |

9. Ex:
Use Simpson’s Rule with = to approximate it.

10. Ex:
Use Simpson’s Rule with = to approximate it.
[ ( )+ ( . )+ ( . ) + · · · + ( )]

11. Ex:
Use Simpson’s Rule with = to approximate it.
[ ( )+ ( . )+ ( . ) + · · · + ( )]
. . . . . .
= [ + + + + +
. . . .
+ + + + + ]

12. Ex:
Use Simpson’s Rule with = to approximate it.
[ ( )+ ( . )+ ( . ) + · · · + ( )]
. . . . . .
= [ + + + + +
. . . .
+ + + + + ]
.

13. Ex:
Estimate the error involved.

14. Ex:
Estimate the error involved.
( )=( + + )

15. Ex:
Estimate the error involved.
( )=( + + )
When
( )

16. Ex:
Estimate the error involved.
( )=( + + )
When
( )
so the error is bounded by
·
.
·

17. Ex:
Estimate the error involved.
( )=( + + )
When
( )
so the error is bounded by
·
.
·
Therefore . ± .