S = \{0 \leq \phi \leq 2\pi, \ 0 \leq \theta \leq \pi, \ 0 \leq \rho \leq R\}
\begin{align*}
\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 \Bigr|_0^{2\pi}\ (-\cos\theta) \Bigr|_0^{ \pi }\ \tfrac13 \rho^3 \Bigr|_0^R \\
&=2\pi \times 2 \times \tfrac13 R^3 \\
&=\tfrac43 \pi R^3
\end{align*}