sphere_volume.tex

{S}=\{ 0\leq {\phi }\leq 2{\pi }, 0\leq {\theta }\leq {\pi }, 0\leq {\rho }\leq {R}\} \begin{array}{rl}
\mathrm{Volume} & =\underset{{S}}{\iiint } \!{{\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 }{\Bigr{\left\mid  }}_{0}^{2{\pi }} (-\cos{\theta }){\Bigr{\left\mid  }}_{0}^{{\pi }} \tfrac{1}{3} {{\rho }}^{3}{\Bigr{\left\mid  }}_{0}^{{R}} \\
 & =2{\pi }\times 2\times \tfrac{1}{3} {{R}}^{3} \\
 & =\tfrac{4}{3} {\pi }{{R}}^{3} \\
\end{array}