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Calculus II - 7

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Stewart Calculus Section 7.7

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Calculus II - 7

1. 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. 2. Basic Idea: Use the summation of area insubintervals to approximate the integral.
3. 3. Basic Idea: Use the summation of area insubintervals to approximate the integral. Midpoint Rule Trapezoidal Rule Simpson’s Rule
4. 4. Midpoint Rule: ( ) [ (¯ ) + (¯ ) + · · · + (¯ )] = ¯ = ( + )
5. 5. Trapezoidal Rule:( ) [ ( )+ ( ) + ··· + ( )+ ( )] =
6. 6. Simpson’s Rule:( ) [ ( )+ ( )+ ( )+ ( ) + ··· + ( )+ ( )+ ( )] = n is e ven!
7. 7. Midpoint Rule: ( ) [ (¯ ) + (¯ ) + · · · + (¯ )] ¯ = ( + )Trapezoidal Rule: ( ) [ ( )+ ( ) + ··· + ( )+ ( )]Simpson’s Rule: ( ) [ ( )+ ( )+ ( )+ ( ) + ··· + ( )+ ( )+ ( )]
8. 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. 9. Ex:Use Simpson’s Rule with = to approximate it.
10. 10. Ex:Use Simpson’s Rule with = to approximate it. [ ( )+ ( . )+ ( . ) + · · · + ( )]
11. 11. Ex:Use Simpson’s Rule with = to approximate it. [ ( )+ ( . )+ ( . ) + · · · + ( )] . . . . . . = [ + + + + + . . . . + + + + + ]
12. 12. Ex:Use Simpson’s Rule with = to approximate it. [ ( )+ ( . )+ ( . ) + · · · + ( )] . . . . . . = [ + + + + + . . . . + + + + + ] .
13. 13. Ex:Estimate the error involved.
14. 14. Ex:Estimate the error involved. ( )=( + + )
15. 15. Ex:Estimate the error involved. ( )=( + + )When ( )
16. 16. Ex:Estimate the error involved. ( )=( + + )When ( )so the error is bounded by · . ·
17. 17. Ex:Estimate the error involved. ( )=( + + )When ( )so the error is bounded by · . ·Therefore . ± .