Surface integral of cylinder

Question

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.

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I'm not sure how to approach this. Any hints?

Answer 1

Surface area $\iint \|dS\|$

what is $dS$?

$(\frac {\partial x}{\partial u},\frac{\partial y}{\partial u},\frac{\partial z}{\partial u})\times (\frac {\partial x}{\partial v},\frac{\partial y}{\partial v},\frac{\partial z}{\partial v}) \ du \ dv$