Volume of a body bounded by a surface.
In spherical coordinates, your equality becomes$$\rho^4=a\rho\cos(\varphi)\rho^2\sin^2(\varphi)=a\rho^3\cos(\varphi)\sin^2(\varphi).$$So, your volume is\begin{align}\int_0^{2\pi}\int_0^{\pi/2}\int_0^{a\cos(\varphi)\sin^2(\varphi)}\rho^2\sin(\varphi)\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta&=2\pi\int_0^{\pi/2}\frac13a^3\cos^3(\varphi)\sin^7(\varphi)\,\mathrm d\varphi\\&=\frac{a^3\pi}{60}.\end{align}