${\int }_0^{2 \mathrm{\pi }}{\int }_0^{{\cos }^{-1}\left(\frac{3}{4}\right)}{\int }_{\frac{3}{4 \cos \left(\mathrm{\phi }\right)}}^1{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right) \mathit{\ⅆ}\mathrm{\rho } \mathit{\ⅆ}\mathrm{\phi } \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{11}{192}\mathrm{\π}$$$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Spherical

Table 7.6.1(a)   Solution by visualization task template

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

Table 7.6.1(b)   Solution by task template that embodies the MultiInt command

Initialize

Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\arccos \left(\frac{3}{4}\right)}{\∫}_{\frac{3}{4\cos \left(\mathrm{\φ}\right)}}^1{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}$$\=$$\frac{11}{192}\mathrm{\π}$

${\int }_0^{2 \mathrm{\π}}{\int }_0^{{\cos }^{-1}\left(\frac{3}{4}\right)}{\int }_{\frac{3}{4 \cos \left(\mathrm{\φ}\right)}}^1{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) \mathit{\ⅆ}\mathrm{\ρ} \mathit{\ⅆ}\mathrm{\φ} \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{11}{192}\mathrm{\π}$$$

Table 7.6.1(c)   Solution from first principles

Initialize

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

Apply the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(1\,\mathrm{\ρ}\=\frac{3}{4 \cos \left(\mathrm{\φ}\right)}..1\,\mathrm{\φ}\=0..{\cos }^{-1}\left(\frac{3}{4}\right)\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\arccos \left(\frac{3}{4}\right)}{\∫}_{\frac{3}{4\cos \left(\mathrm{\φ}\right)}}^1{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}$

$\mathrm{MultiInt}\left(1\,\mathrm{\rho }\=\frac{3}{4 \cos \left(\mathrm{\phi }\right)}..1\,\mathrm{\phi }\=0..{\cos }^{-1}\left(\frac{3}{4}\right)\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)$

$\frac{11}{192}\mathrm{\π}$

$\mathrm{MultiInt}\left(1\,\mathrm{\rho }\=\frac{3}{4 \cos \left(\mathrm{\phi }\right)}..1\,\mathrm{\phi }\=0..{\cos }^{-1}\left(\frac{3}{4}\right)\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\,\mathrm{output}\=\mathrm{steps}\right)$

$\begin{array}{ccc}\multicolumn{3}{c}{{\int }_0^{2\mathrm{\pi }}{\int }_0^{\arccos \left(\frac{3}{4}\right)}{\int }_{\frac{3}{4\cos \left(\mathrm{\phi }\right)}}^1{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)\ⅆ\mathrm{\rho }\ⅆ\mathrm{\phi }\ⅆ\mathrm{\theta }}\\ & \text{=}& {\int }_0^{2\mathrm{\pi }}{\int }_0^{\arccos \left(\frac{3}{4}\right)}\left(\genfrac{}{}{0ex}{}{\frac{{\mathrm{\rho }}^3\sin \left(\mathrm{\phi }\right)}{3}}{\phantom{\mathrm{\rho }=\frac{3}{4\cos \left(\mathrm{\phi }\right)}..1}}|\genfrac{}{}{0ex}{}{\phantom{\frac{{\mathrm{\rho }}^3\sin \left(\mathrm{\phi }\right)}{3}}}{\mathrm{\rho }=\frac{3}{4\cos \left(\mathrm{\phi }\right)}..1}\right)\ⅆ\mathrm{\phi }\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}{\int }_0^{\arccos \left(\frac{3}{4}\right)}\frac{\sin \left(\mathrm{\phi }\right)\left(1-\frac{27}{64{\cos \left(\mathrm{\phi }\right)}^3}\right)}{3}\ⅆ\mathrm{\phi }\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}\left(\genfrac{}{}{0ex}{}{\left(-\frac{\cos \left(\mathrm{\phi }\right)}{3}-\frac{9}{128{\cos \left(\mathrm{\phi }\right)}^2}\right)}{\phantom{\mathrm{\phi }=0..\arccos \left(\frac{3}{4}\right)}}|\genfrac{}{}{0ex}{}{\phantom{\left(-\frac{\cos \left(\mathrm{\phi }\right)}{3}-\frac{9}{128{\cos \left(\mathrm{\phi }\right)}^2}\right)}}{\mathrm{\phi }=0..\arccos \left(\frac{3}{4}\right)}\right)\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& {\int }_0^{2\mathrm{\pi }}\frac{11}{384}\ⅆ\mathrm{\theta }\hfill \\ & \text{=}& \genfrac{}{}{0ex}{}{\frac{11\mathrm{\theta }}{384}}{\phantom{\mathrm{\theta }=0..2\mathrm{\pi }}}|\genfrac{}{}{0ex}{}{\phantom{\frac{11\mathrm{\theta }}{384}}}{\mathrm{\theta }=0..2\mathrm{\pi }}\hfill \end{array}$

Solution via the top-level Int and int commands

$$\begin{gathered} \mathrm{Int}\left({\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\,\mathrm{\ρ}\=\frac{3}{4 \cos \left(\mathrm{\φ}\right)}..1\,\mathrm{\φ}\=0..{\cos }^{-1}\left(\frac{3}{4}\right)\,\mathrm{\θ}\=0..2 \mathrm{\π}\right)\= \\ \mathrm{int}\left({\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)\,\mathrm{\rho }\=\frac{3}{4 \cos \left(\mathrm{\phi }\right)}..1\,\mathrm{\phi }\=0..{\cos }^{-1}\left(\frac{3}{4}\right)\,\mathrm{\theta }\=0..2 \mathrm{\pi }\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\arccos \left(\frac{3}{4}\right)}{\∫}_{\frac{3}{4\cos \left(\mathrm{\φ}\right)}}^1{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}\=\frac{11}{192}\mathrm{\π}$