= Solution
{c}
For a regular <parametrized surface> that covers its image once, up to sets of zero <area>, its <surface area> is
$$
\boxed{\operatorname{Area}(S)=\iint_U\left\|\frac{\partial\mathbf r}{\partial u}\times\frac{\partial\mathbf r}{\partial v}\right\|\,du\,dv.}
$$
The integrand is the <area element of a surface>, the <area> of the small <parallelogram> spanned by the two <tangent vectors>. A <parametrization> that covers a region more than once counts that multiplicity; the geometric <surface area> requires a single covering or a division into nonoverlapping patches.
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