= Solution
The <polar area element> is $r\,dr\,d\phi$. Integrating first in <radius> and using symmetry gives
$$
\begin{aligned}
\mathcal A
&=\frac12\int_{-\pi/3}^{\pi/3}\left[(1+\cos\phi)^2-\frac94\right]d\phi\\
&=\int_0^{\pi/3}\left[-\frac34+2\cos\phi+\frac12\cos2\phi\right]d\phi\\
&=\left[-\frac{3\phi}{4}+2\sin\phi+\frac14\sin2\phi\right]_0^{\pi/3}
=\boxed{\frac{9\sqrt3}{8}-\frac\pi4.}
\end{aligned}
$$
This is the specialization $a=1$, $c=3/2$ of the <polar cap between a cardioid and a circle> area formula.
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