Calculus II - 7
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This document discusses numerical integration techniques for approximating integrals of functions that have no closed-form antiderivative. It introduces the midpoint rule, trapezoidal rule, and Simpson's rule for approximating integrals using Riemann sums over subintervals. An example applies Simpson's rule to approximate a definite integral and estimates the error bound.
![Midpoint Rule:
( ) [ (¯ ) + (¯ ) + · · · + (¯ )]
= ¯ = ( + )](https://image.slidesharecdn.com/math12722011-09-21-111203211855-phpapp01/85/Calculus-II-7-4-320.jpg)
![Trapezoidal Rule:
( ) [ ( )+ ( ) + ··· + ( )+ ( )]
=](https://image.slidesharecdn.com/math12722011-09-21-111203211855-phpapp01/85/Calculus-II-7-5-320.jpg)
![Simpson’s Rule:
( ) [ ( )+ ( )+ ( )+ ( ) + ···
+ ( )+ ( )+ ( )]
= n is e
ven!](https://image.slidesharecdn.com/math12722011-09-21-111203211855-phpapp01/85/Calculus-II-7-6-320.jpg)
![Midpoint Rule:
( ) [ (¯ ) + (¯ ) + · · · + (¯ )]
¯ = ( + )
Trapezoidal Rule:
( ) [ ( )+ ( ) + ··· + ( )+ ( )]
Simpson’s Rule:
( ) [ ( )+ ( )+ ( )+ ( ) + ···
+ ( )+ ( )+ ( )]](https://image.slidesharecdn.com/math12722011-09-21-111203211855-phpapp01/85/Calculus-II-7-7-320.jpg)
![Ex:
Use Simpson’s Rule with = to approximate it.
[ ( )+ ( . )+ ( . ) + · · · + ( )]](https://image.slidesharecdn.com/math12722011-09-21-111203211855-phpapp01/85/Calculus-II-7-10-320.jpg)
![Ex:
Use Simpson’s Rule with = to approximate it.
[ ( )+ ( . )+ ( . ) + · · · + ( )]
. . . . . .
= [ + + + + +
. . . .
+ + + + + ]](https://image.slidesharecdn.com/math12722011-09-21-111203211855-phpapp01/85/Calculus-II-7-11-320.jpg)
![Ex:
Use Simpson’s Rule with = to approximate it.
[ ( )+ ( . )+ ( . ) + · · · + ( )]
. . . . . .
= [ + + + + +
. . . .
+ + + + + ]
.](https://image.slidesharecdn.com/math12722011-09-21-111203211855-phpapp01/85/Calculus-II-7-12-320.jpg)
Calculus II - 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 . ± .