= Solution
{c}
In <polar coordinates>, put $x=r\cos\theta$ and $y=r\sin\theta$. Differentiating along the curve gives
$$
x\,dy-y\,dx=r^2\,d\theta.
$$
The <cardioid> has $r\geq0$ and is traversed once counterclockwise as $\theta$ increases from $0$ to $2\pi$. Its cusp at $\theta=\pi$ causes no difficulty: split the <line integral> there, or exclude a small arc and take its limit. Thus <signed area as a boundary integral> gives
$$
\begin{aligned}
|D|&=\frac12\int_0^{2\pi}(1+\cos\theta)^2\,d\theta\\
&=\frac12\int_0^{2\pi}\left(\frac32+2\cos\theta+\frac12\cos2\theta\right)d\theta
=\boxed{\frac{3\pi}{2}}.
\end{aligned}
$$
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