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Math Help - Riemann-Stieltjes Integral

  1. #1
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    Riemann-Stieltjes Integral

    Hello!

    I am working on this problem:

    Suppose f and g are continuous and positive on some interval [a,b].

    I am trying to show that \exists\  \zeta \in [a,b] such that \int_a^bf(x)g(x)dx = f(\zeta)\int_a^bg(x)dx

    All I have so far is that since both f and g are continuous, then their product is continuous (and therefore Riemann-integrable) but I'm not sure how to select this zeta. Whole thing looks like some sort of mean value theorem for products of functions?!

    Any help appreciated. Thanks!!
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  2. #2
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    Quote Originally Posted by matt.qmar View Post
    Hello!

    I am working on this problem:

    Suppose f and g are continuous and positive on some interval [a,b].

    I am trying to show that \exists\  \zeta \in [a,b] such that \int_a^bf(x)g(x)dx = f(\zeta)\int_a^bg(x)dx

    All I have so far is that since both f and g are continuous, then their product is continuous (and therefore Riemann-integrable) but I'm not sure how to select this zeta. Whole thing looks like some sort of mean value theorem for products of functions?!

    Any help appreciated. Thanks!!
    You are correct this is a mean value theorem for integrals

    See this link

    Mean value theorem - Wikipedia, the free encyclopedia

    The proof on the wiki page shows you how to select the value you wish.
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