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Alternative proof for the 1st mean-value theorem for integrals

  1. Sep 21, 2009 #1
    can anyone tell me how to prove the 1st mean-value theorem for integral
    by applying Lagrange mean-value theorem to an integral with variable upper limit?
    thanks a lot.
  2. jcsd
  3. Sep 21, 2009 #2
    I think I've seen a proof that uses the Cauchy Mean Value Theorem using

    [tex]F(t) = \int_{a}^{t}f(x)g(x)\;dx[/tex]

    [tex]G(t) = \int_{a}^{t}g(x)\;dx[/tex]

    So there exists [itex]\xi[/itex] so that

    [tex]\frac{F'(\xi)}{G'(\xi)} = \frac{F(b)-F(a)}{G(b)-G(a)}[/tex]

    or at least that is what I remember.

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