= Solution
{c}
An infinitesimal parameter rectangle becomes a <parallelogram> with edges $\mathbf r_s\,ds$ and $\mathbf r_t\,dt$. Its area is the <Euclidean norm> of their <cross product>, giving the <surface area element>
$$
dS=|\mathbf r_s\times\mathbf r_t|\,ds\,dt.
$$
Thus, for a regular <parametrization> covering the <surface> once except possibly on sets of area zero,
$$
\boxed{\operatorname{area}(S)=\iint_U|\mathbf r_s\times\mathbf r_t|\,ds\,dt.}
$$
For several patches, add their <surface integrals> while avoiding overlap; repeated coverage would otherwise count area with multiplicity.
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