Volume enclosed by a sphere ($V=4\pi a^3/3$)

The solid ball inside a radius-$a$ sphere has volume $4\pi a^3/3$. One derivation integrates concentric shell area: $V={\int }_0^{a}4\pi r^{2}dr$. The sphere itself is the boundary surface; its area is $4\pi a^2$, a different dimensional quantity.