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Simpson's Rule/Trapezoidal Approximation - Error rate help

  1. May 5, 2010 #1
    1. The problem statement, all variables and given/known data

    [tex] \int^{ \pi}_{0} sin(x)dx \;\;\;\;\;\;\;\; dx=\frac{ \pi}{2}[/tex]

    2. Relevant equations

    Trapezoidal Approximation:

    [tex]|f''(x)| \leq M \;\;\;\;\; for \;\;\;\;\; a \leq x \leq b [/tex]

    [tex] \frac {b-a}{12}(M)(dx)^{2} = Error [/tex]

    Simpson's Rule:

    [tex] |f^{(4)}(x)| \leq M \;\;\;\;\; for \;\;\;\;\; a \leq x \leq b [/tex]

    [tex] \frac{b-a}{180}(M)(dx)^{4} = Error [/tex]

    3. The attempt at a solution

    Ok so I have found the correct estimations using both methods easily, the trapezoidal approximation is: 1.5708 and Simpson's Rule is: 2.0944, those numbers check out in the back of the book, but when it comes to finding the error I think that it should be 0 because the max (M) is zero for both the second and fourth derivative but the book says otherwise. Heres what I did:

    [tex] y=sin(x) [/tex]

    [tex] y'=cos(x) [/tex]

    [tex] y''=-sin(x) [/tex]

    [tex] y^{(3)}= -cos(x) [/tex]

    [tex] y^{(4)}= sin(x) [/tex]

    and Trapezoidal Rule using y'' is:

    [tex]|y''( \pi)|=0 [/tex]

    and

    [tex] |y''(0)|=0 [/tex]

    and that follows the same for [tex] y^{(4)} [/tex] so M is 0 and thus the entire equation is 0 and Error = 0 but the book states that the error for the trapezoidal approximation is:

    [tex] \frac { \pi^{3}}{48} \;\;\;\;\; or \;\;\;\; .65 [/tex]

    and the Error for Simpson's Rule is:

    [tex] \frac { \pi^{5}}{2880} \;\;\;\; or \;\;\;\; .1 [/tex]

    I don't see how they got this.... but I don't think the actual error rate is zero either because if it were then the trapezoidal and simpson approximation would be exactly equal, so where did I go wrong?
     
  2. jcsd
  3. May 5, 2010 #2

    Mark44

    Staff: Mentor

    f''(x) = -sin(x) and f(4)(x) = sin(x). The maximum value of the absolute values of these functions is not zero. What you're looking for is the maximum value over the entire interval [0, pi].
     
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