Polar cap between a cardioid and a circle ($\mathcal{A}=a^2(3\beta /2+2\sin \beta +\sin 2\beta /4)-c^2\beta$)

For $a>0$ and $0<c<2a$, consider the region $c\le r\le a(1+\cos \varphi )$ in polar coordinates. It exists precisely for $|\varphi |\le \beta$, where $\beta =\arccos (c/a-1)\in (0,\pi )$. The polar area element $rdrd\varphi$ givesExpanding 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.