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Math Help - Area between sinh(x) and cosh(x)

  1. #1
    Junior Member Greg98's Avatar
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    Area between sinh(x) and cosh(x)

    Hello,
    the problem is to calculate area between functions sinh(x) and cosh(x) on first quarter. Picture illustrates task better:

    Area between sinh(x) and cosh(x)-functions.jpg

    I tried as follows:
    <br />
A = \int_{0}^{\infty} cosh(x) dx - \int_{0}^{\infty} sinh(x) dx

    \int_{0}^{\infty} cosh(x) dx
    = lim_{\beta \to \infty} \int_{0}^{\beta} cosh(x) dx
    = sinh(\beta) - sinh(0)
    = sinh(\beta) \rightarrow \infty , when \ \beta \rightarrow infty

    \int_{0}^{\infty} sinh(x) dx
    = lim_{\beta \to \infty} \int_{0}^{\beta} sinh(x) dx
    = cosh(\beta) - cosh(0)
    = cosh(\beta) - 1 \rightarrow\infty , when \ \beta \rightarrow infty

    So, I get \infty - \infty , which isn't correct answer, I think. Am I right or am I right? I don't know how to calculate it better, so any help is appreciated. Thanks very much!
    Last edited by Greg98; November 18th 2010 at 07:23 AM.
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  2. #2
    MHF Contributor
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    Note that \cosh x-\sinh x=e^{-x}.
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