sphere_volume.tex

S = \{ 0 \leq \phi \leq 2\pi,\ 0 \leq \theta \leq \pi,\ 0 \leq \rho \leq R\}\begin{aligned}
\mathrm{Volume} & = \iiint\limits_{S}\!\rho^{2}\sin\theta\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\phi \\
 & = \int_{0}^{2\pi}\!\mathrm{d}\phi\,\int_{0}^{\pi}\!\sin\theta\,\mathrm{d}\theta\,\int_{0}^{R}\!\rho^{2}\mathrm{d}\rho \\
 & = \phi\Big|_{0}^{2\pi}\ ( - \cos\theta)\Big|_{0}^{\pi}\ \tfrac{1}{3}\rho^{3}\Big|_{0}^{R} \\
 & = 2\pi \times 2 \times \tfrac{1}{3}R^{3} \\
 & = \tfrac{4}{3}\pi R^{3} \\
\end{aligned}