Thread: Area between sinh(x) and cosh(x)

1. 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:

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!

2. Note that $\displaystyle \cosh x-\sinh x=e^{-x}$.