= Solution
The radial interval is nonempty exactly when $1+\cos\phi\geq3/2$, or $\cos\phi\geq1/2$. Choose the representative angular range $-\pi/3\leq\phi\leq\pi/3$. The <region of integration> is therefore
$$
\boxed{-\frac\pi3\leq\phi\leq\frac\pi3,\qquad
\frac32\leq r\leq1+\cos\phi.}
$$
It is the right-hand cap inside the <cardioid> and outside the <circle> of <radius> $3/2$, symmetric about the horizontal axis. Its boundary intersections have coordinates $(3/4,\pm3\sqrt3/4)$, and the cap reaches $(2,0)$. The radial segment on the horizontal axis runs from $3/2$ to $2$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1-cardioid-cap.png]
{title=The integration cap inside a cardioid and outside the concentric circle}
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