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Thread: 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 $\displaystyle sinh(x)$ and $\displaystyle cosh(x)$ on first quarter. Picture illustrates task better:

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

    I tried as follows:
    $\displaystyle
    A = \int_{0}^{\infty} cosh(x) dx - \int_{0}^{\infty} sinh(x) dx$

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

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

    So, I get $\displaystyle \infty - \infty$ , 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; Nov 18th 2010 at 07:23 AM.
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  2. #2
    MHF Contributor
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    Note that $\displaystyle \cosh x-\sinh x=e^{-x}$.
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