= Polar cap between a cardioid and a circle
{title2=$\mathcal A=a^2(3\beta/2+2\sin\beta+\sin2\beta/4)-c^2\beta$}
For $a>0$ and $0<c<2a$, consider the region $c\leq r\leq a(1+\cos\phi)$ in <polar coordinates>. It exists precisely for $|\phi|\leq\beta$, where $\beta=\arccos(c/a-1)\in(0,\pi)$. The <polar area element> $r\,dr\,d\phi$ gives
$$
\mathcal A=\frac12\int_{-\beta}^{\beta}\big[a^2(1+\cos\phi)^2-c^2\big]d\phi.
$$
Expanding the square and integrating its even terms gives the title formula. This cap lies inside the <cardioid> and outside the concentric <circle>, so reversing the radial bounds would select the wrong region or a signed area.
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