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Math Help - Simpson's rule

  1. #1
    Junior Member
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    Simpson's rule

    The Trapezoid rule applied to \int_0^2 \! f(x) \, dx. gives the value 5 and the Midpoint rule gives the value 4. What value does Simpson's rule give?

    It should be 13/3 but I don't see how to get it.
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  2. #2
    Super Member Deadstar's Avatar
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    Midpoint Rule => M = (b-a)f \left (\frac{a+b}{2} \right )

    Trapezoidal Rule => T = \frac{1}{2}(b-a)(f(a) + f(b))

    Simpsons Rule => S = \frac{b-a}{6} \bigg{[}f(a) + 4f \left (\frac{a+b}{2} \right ) + f(b) \bigg{]} = \frac{(b-a)(f(a) + f(b))}{6} + \frac{2}{3}(b-a)f \left (\frac{a+b}{2} \right )

    = \frac{T}{3} + \frac{2M}{3} = \frac{5}{3} + \frac{8}{3} = \frac{13}{3}
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