$$S=\{0\le \varphi \le 2\pi ,0\le \theta \le \pi ,0\le \rho \le R\}\begin{array}{cc}\hfill Volume& =\underset{S}{\iiint }{\rho }^{2}sin\theta d\rho d\theta d\varphi \hfill \\ \hfill & ={\int }_0^{2\pi }d\varphi {\int }_0^{\pi }sin\theta d\theta {\int }_0^R{\rho }^{2}d\rho \hfill \\ \hfill & =\varphi {|}_0^{2\pi }(-cos\theta ){|}_0^{\pi }\frac{1}{3}{\rho }^3{|}_0^R\hfill \\ \hfill & =2\pi \times 2\times \frac{1}{3}{R}^3\hfill \\ \hfill & =\frac{4}{3}\pi R^3\hfill \end{array}$$