= Solution
The <cardioid> has nonnegative polar radius $r=1-\cos\theta$, ranging from zero at the right-hand cusp to two on the negative $x$ axis. Its intersections with the circle $r=3/2$ satisfy $\cos\theta=-1/2$, giving $\theta=2\pi/3,4\pi/3$. The selected region is the left-hand cap between the two curves:
$$
\boxed{\frac{2\pi}{3}\leq\theta\leq\frac{4\pi}{3},\qquad\frac32\leq r\leq1-\cos\theta.}
$$
The endpoints in <Cartesian coordinates> are $(-3/4,\pm3\sqrt3/4)$.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013-cardioid-exterior-circle-region.png]
{title=The cardioid cap outside the circle and its radial area integrand}
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