$\mathrm{\rho }$

$\mathrm{\ρ}$

$0$

$0$

$1$

$1$

$0$

$0$

$\frac{\mathrm{\π}}{6}$

$\frac{1}{6}\mathrm{\π}$

$0$

$0$

$2 \mathrm{\pi }$

$2\mathrm{\π}$

$\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(\,\mathrm{\ρ}\=..\,\mathrm{\φ}\=..\,\mathrm{\θ}\=..\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\theta }\right]\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{6}\mathrm{\π}}{\∫}_0^1{\mathrm{\ρ}}^3\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}$

$\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(\,\mathrm{\ρ}\=..\,\mathrm{\φ}\=..\,\mathrm{\θ}\=..\, \mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\theta }\right]\right)$

$\frac{1}{2}\mathrm{\π}-\frac{1}{4}\sqrt{3}\mathrm{\π}$

$\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(\,\mathrm{\ρ}\=..\, \mathrm{\φ}\=..\,\mathrm{\θ}\=..\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\theta }\right]\,\mathrm{output}\=\mathrm{steps}\right)$

$\begin{array}{ccc}\multicolumn{3}{c}{{\int }_0^{2\mathrm{\pi }}{\int }_0^{\frac{\mathrm{\pi }}{6}}{\int }_0^1{\mathrm{\rho }}^3\sin \left(\mathrm{\phi }\right)\ⅆ\mathrm{\rho }\ⅆ\mathrm{\phi }\ⅆ\mathrm{\theta }}\\ & \text{=}& {\int }_0^{2\mathrm{\pi }}{\int }_0^{\frac{\mathrm{\pi }}{6}}\left(\genfrac{}{}{0ex}{}{\frac{{\mathrm{\rho }}^4\sin \left(\mathrm{\phi }\right)}{4}}{\phantom{\mathrm{\rho }=0..1}}|\genfrac{}{}{0ex}{}{\phantom{\frac{{\mathrm{\rho }}^4\sin \left(\mathrm{\phi }\right)}{4}}}{\mathrm{\rho }=0..1}\right)\ⅆ\mathrm{\phi }\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}{\int }_0^{\frac{\mathrm{\pi }}{6}}\frac{\sin \left(\mathrm{\phi }\right)}{4}\ⅆ\mathrm{\phi }\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}\left(\genfrac{}{}{0ex}{}{-\frac{\cos \left(\mathrm{\phi }\right)}{4}}{\phantom{\mathrm{\phi }=0..\frac{\mathrm{\pi }}{6}}}|\genfrac{}{}{0ex}{}{\phantom{-\frac{\cos \left(\mathrm{\phi }\right)}{4}}}{\mathrm{\phi }=0..\frac{\mathrm{\pi }}{6}}\right)\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}\left(\frac{1}{4}-\frac{\sqrt{3}}{8}\right)\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& \genfrac{}{}{0ex}{}{\left(\frac{1}{4}-\frac{\sqrt{3}}{8}\right)\mathrm{\theta }}{\phantom{\mathrm{\theta }=0..2\mathrm{\pi }}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{1}{4}-\frac{\sqrt{3}}{8}\right)\mathrm{\theta }}}{\mathrm{\theta }=0..2\mathrm{\pi }}\hfill \end{array}$

$\frac{1}{2}\mathrm{\π}-\frac{1}{4}\sqrt{3}\mathrm{\π}$