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Math Help - Integral problem involving the definition of the integral

  1. #1
    Junior Member
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    Integral problem involving the definition of the integral

    Hello

    I've been studying Darboux integrals, and I can't solve the following problem: Pn is the set of all equally spaced partitions of [0,1] such that Pn = {0, 1/n, 2/n, ... 1}
    f(x) = e^x
    I'm trying to prove that the integral from 0 to 1 of e^x = sup{L(f,Pn)}
    In other words that the integral is equal to the supremum of the set of lower sums of f on equally spaced partitions.
    From Darboux's definition of the integral, I know that the integral is equal to the supremum of the lower sums over ALL partitions. How can I limit that to just equally spaced partitions?
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  2. #2
    Senior Member Tinyboss's Avatar
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    Choose an arbitrary partition P, and epsilon positive. Show that there is an equally-spaced partition E such that L(e^x,E)+epsilon > L(e^x,P).
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