= Solution
In <polar coordinates>, the area element is $r\,dr\,d\theta$. Using the bounds from part (a),
$$
\mathcal A=\frac12\int_{2\pi/3}^{4\pi/3}\left[(1-\cos\theta)^2-\frac94\right]d\theta.
$$
Since the bracket equals $-3/4-2\cos\theta+(1/2)\cos2\theta$, an antiderivative is $-3\theta/4-2\sin\theta+(1/4)\sin2\theta$. Evaluation gives
$$
\boxed{\mathcal A=\frac{9\sqrt3}{8}-\frac\pi4.}
$$
The inner-circle contribution must be subtracted before angular integration; integrating the entire cardioid or the entire circle would include regions excluded by the specified intersection.
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