I want to calculate the area, length of curve and volume of $f(x,y) = \cosh(x)$ in a domain $D = \{(x,y), \in \mathbb{R}^2: 0\leq x\leq 1, |y|\leq \sinh(x)\}$
I want to calculate (a) the area of $S$ which is induced by $f(x,y)$, (b) the length of the border curve of $S$ and (c) the volumne between $D$ and $S$
For (a) I thought of simply integrating $$\int_0^1 \int_{-\sinh(x)}^{\sinh(x)} \cosh(x) dydx = ...= \sinh(1)^2$$
For (b) I thought of getting a parametrisation $\gamma$ of $D$ and then using $$\int_{\partial D} f ds = \int_0^1 f(\gamma(t)) ||\gamma'(t)||_2 dt$$
but I can't really think of a parametrisation. Can $\cosh(x) = \sqrt{1- \sinh(x)^2}$ be of any use here?
For (c) I have no idea sadly, I would really like some tips there.