${\int }_0^{2 \mathrm{\π}}{\int }_0^{\mathrm{\π}}{\int }_0^{ 2 \sin \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)$ $d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$ = $2 {\mathrm{\π}}^2$ 

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Spherical

Table 8.1.26(a)   Task template integrating in spherical coordinates

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Spherical

Table 8.1.26(b)   Task template implementing the MultiInt command iterating in the order $d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$

Initialize

Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{2\sin \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}$$\=$$2{\mathrm{\π}}^2$

${\int }_0^{2 \mathrm{\π}}{\int }_0^{\mathrm{\π}}{\int }_0^{2 \sin \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) \mathit{\ⅆ}\mathrm{\ρ} \mathit{\ⅆ}\mathrm{\φ} \mathit{\ⅆ}\mathrm{\θ}$ = $2{\mathrm{\π}}^2$$$

Table 8.1.26(c)   Integration via first principles

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$\mathrm{MultiInt}\left(1\,\mathrm{\ρ}\=0..2 \sin \left(\mathrm{\φ}\right)\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{integral}\right)$

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

$\mathrm{MultiInt}\left(1\,\mathrm{\rho }\=0..2 \sin \left(\mathrm{\phi }\right)\,\mathrm{\phi }\=0..\mathrm{\pi }\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)$ = $2{\mathrm{\π}}^2$$$

Table 8.1.26(d)   MultiInt command iterating in spherical coordinates in the order $d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$

$$\begin{gathered} \mathrm{Int}\left({\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\,\left[\mathrm{\rho }\=0..2 \sin \left(\mathrm{\φ}\right)\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)\= \\ \mathrm{int}\left({\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\,\left[\mathrm{\rho }\=0..2 \sin \left(\mathrm{\φ}\right)\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{2\sin \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}\=2{\mathrm{\π}}^2$

Table 8.1.21(e)   Top-level Int and int commands