${\int }_0^{2 \mathrm{\π}}{\int }_{\mathrm{\π}\/6}^{\mathrm{\π}\/4}{\int }_0^2{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)$ $d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$ = $\frac{8}{3}\mathrm{\π} \left(\sqrt{3}-\sqrt{2}\right)$ 

$z$

$\= \sqrt{3\left(x^2\+y^2\right)}$

$z^2$

$\= 3\left(x^2\+y^2\right)$

${\left(\mathrm{\ρ} \cos \left(\mathrm{\φ}\right)\right)}^2$

$\= 3\left({\left(\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)\right)}^2\+{\left(\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)\right)}^2\right)$

${\mathrm{\ρ}}^2{\cos }^2\left(\mathrm{\φ}\right)$

$\= 3 {\mathrm{\ρ}}^2\left({\sin }^2\left(\mathrm{\φ}\right) \left({\cos }^2\left(\mathrm{\θ}\right)\+{\sin }^2\left(\mathrm{\θ}\right)\right)\right)$

${\cos }^2\left(\mathrm{\φ}\right)$

$\= 3 {\sin }^2\left(\mathrm{\φ}\right)$

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

Table 8.1.16(a)   Task template integrating in spherical coordinates

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

Table 8.1.16(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}}{\∫}_{\frac{1}{6}\mathrm{\π}}^{\frac{1}{4}\mathrm{\π}}{\∫}_0^2{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}$$\=$$\frac{8}{3}\mathrm{\π}\sqrt{3}-\frac{8}{3}\sqrt{2}\mathrm{\π}$

${\int }_0^{2 \mathrm{\π}}{\int }_{\mathrm{\π}\/6}^{\mathrm{\π}\/4}{\int }_0^2{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) \mathit{\ⅆ}\mathrm{\ρ} \mathit{\ⅆ}\mathrm{\φ} \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{8}{3}\sqrt{3}\mathrm{\π}-\frac{8}{3}\sqrt{2}\mathrm{\π}$$$

Table 8.1.16(c)   Integration via first principles

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

$\mathrm{MultiInt}\left(1\,\mathrm{\ρ}\=0..2\,\mathrm{\φ}\=\mathrm{\π}\/6..\mathrm{\π}\/4\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_{\frac{1}{6}\mathrm{\π}}^{\frac{1}{4}\mathrm{\π}}{\∫}_0^2{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}$

$\mathrm{MultiInt}\left(1\,\mathrm{\rho }\=0..2\,\mathrm{\phi }\=\mathrm{\pi }\/6..\mathrm{\pi }\/4\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)$

$\frac{8}{3}\sqrt{3}\mathrm{\π}-\frac{8}{3}\sqrt{2}\mathrm{\π}$

Table 8.1.16(d)   MultiInt command iterating in spherical coordinates in the order $d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}$

$$\begin{gathered} \mathrm{Int}\left({\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\,\left[\mathrm{\ρ}\=0..2\,\mathrm{\φ}\=\mathrm{\π}\/6..\mathrm{\π}\/4\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)\= \\ \mathrm{int}\left({\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\,\left[\mathrm{\ρ}\=0..2\,\mathrm{\φ}\=\mathrm{\π}\/6..\mathrm{\π}\/4\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_{\frac{1}{6}\mathrm{\π}}^{\frac{1}{4}\mathrm{\π}}{\∫}_0^2{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}\=\frac{16}{3}\left(\frac{1}{2}\sqrt{3}-\frac{1}{2}\sqrt{2}\right)\mathrm{\π}$

Table 8.1.16(e)   Top-level Int and int commands