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Math Help - another question on approximating integrations

  1. #1
    Junior Member
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    another question on approximating integrations

    Let M_n be the nth Midpoint Rule approximation for a function f.
    Let T_n be the nth Trapezoidal Rule approximation.
    Show that if f''(x) >= 0 (i.e. f is convex on [a,b]), then for any natural numbers m,n, we have
    M_n <= integral f(x)dx <= T_m.
    If f''(x) <= 0 on [a,b], this inequality is reversed.

    I've been playing around with the respective error bounds forever. How do you make this work algebraically? I'm having trouble simplifying the expression and working out the absolute values.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by BrainMan View Post
    Let M_n be the nth Midpoint Rule approximation for a function f.
    Let T_n be the nth Trapezoidal Rule approximation.
    Show that if f''(x) >= 0 (i.e. f is convex on [a,b]), then for any natural numbers m,n, we have
    M_n <= integral f(x)dx <= T_m.
    If f''(x) <= 0 on [a,b], this inequality is reversed.

    I've been playing around with the respective error bounds forever. How do you make this work algebraically? I'm having trouble simplifying the expression and working out the absolute values.
    Draw a diagram showing just one of the intervals and the area corresponding
    to each method's contribution to it's estimate of the integral.
    Compare these with the actual area under the curve on the interval.

    Now turn that into an analyitc proof.

    RonL
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  3. #3
    Junior Member
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    Can you use the error inequalities? I'm trying that but I can't equate the different bounds and I get the integral times 2.

    Thanks for answering.
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