7.7 Approximate Integration   Some functions have no explicit anti-   derivative, e.g. −     ,      and so on.   In practi...
Basic Idea: Use the summation of area insubintervals to approximate the integral.
Basic Idea: Use the summation of area insubintervals to approximate the integral.    Midpoint Rule    Trapezoidal Rule    ...
Midpoint Rule:    ( )          [ (¯ ) + (¯ ) + · · · + (¯ )]          =            ¯ =    (     + )
Trapezoidal Rule:( )          [ ( )+   ( ) + ··· +   (   )+ (   )]         =
Simpson’s Rule:( )         [ ( )+   ( )+    ( )+       ( ) + ···                     +   (   )+     (      )+ (      )]   ...
Midpoint Rule:    ( )           [ (¯ ) + (¯ ) + · · · + (¯ )]                                           ¯ =        (    + ...
Error Bound of Midpoint Rule:Suppose |     ( )|        for                                  (   )              then    |  ...
Ex:Use Simpson’s Rule with   =   to approximate it.
Ex:Use Simpson’s Rule with    =       to approximate it.              [ ( )+      ( . )+    ( . ) + · · · + ( )]
Ex:Use Simpson’s Rule with             =       to approximate it.                      [ ( )+    ( . )+       ( . ) + · · ...
Ex:Use Simpson’s Rule with             =       to approximate it.                      [ ( )+    ( . )+       ( . ) + · · ...
Ex:Estimate the error involved.
Ex:Estimate the error involved.           ( )=(    +          +   )
Ex:Estimate the error involved.           ( )=(    +          +   )When                     ( )
Ex:Estimate the error involved.           ( )=(           +         +   )When                           ( )so the error is...
Ex:Estimate the error involved.            ( )=(           +           +    )When                            ( )so the err...
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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 . ± .

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