A cylinder surface in $(x,y,z)$-space is given by the parametric form:
$$ \begin{bmatrix} x \\ y \\ z \end{bmatrix}=r(u,v)=\begin{bmatrix} \exp(u)+\exp(-u) \\ 2u \\ v(\exp(u)-\exp(-u)) \end{bmatrix}, 0 \leq u \leq 1, 0 \leq v \leq 1$$
Determine its area.
I'm not sure how to approach this. Any hints?
Surface area $\iint \|dS\|$
what is $dS$?
$(\frac {\partial x}{\partial u},\frac{\partial y}{\partial u},\frac{\partial z}{\partial u})\times (\frac {\partial x}{\partial v},\frac{\partial y}{\partial v},\frac{\partial z}{\partial v}) \ du \ dv$