= Solution
{c}
The inequalities become the <rectangle> $1/2\leq u\leq1$, $1/2\leq v\leq1$ in <parabolic polar coordinates>. The <change of variables formula> and the <Jacobian determinant> from <a> give the <area>
$$
\begin{aligned}
A&=\int_{1/2}^1\int_{1/2}^1(u^2+v^2)\,dv\,du\\
&=\frac12\int_{1/2}^1u^2\,du+\frac12\int_{1/2}^1v^2\,dv
=\left[\frac{s^3}{3}\right]_{1/2}^1
=\boxed{\frac7{24}}.
\end{aligned}
$$
The middle panel in <b> shows the curved region in <Cartesian coordinates>; its straight edges in <parabolic polar coordinates> explain the simple <integration limits>.
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