= Solution
{c}
Let $D$ be a bounded planar region with a piecewise smooth, simple closed boundary $C$, oriented counterclockwise, and let $P,Q$ have continuous first <partial derivatives> on an open neighbourhood of its closure. <Green's theorem> states that
$$
\boxed{\oint_C(P\,dx+Q\,dy)=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dx\,dy.}
$$
For a region with holes, orient every boundary component with $D$ on its left: the outer component is counterclockwise and the inner components are clockwise.
Choose $P=-y/2$ and $Q=x/2$. Their indicated <partial derivatives> differ by one, so the <area> is
$$
\boxed{|D|=\iint_D1\,dx\,dy=\frac12\oint_C(x\,dy-y\,dx).}
$$
This is <signed area as a boundary integral>; reversing the direction of $C$ changes the sign of the integral.
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