$\frac{{\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{2\sin \left(\mathrm{\phi }\right)}{\mathrm{\rho }}^{7\/2}\sin \left(\mathrm{\phi }\right) d\mathrm{\rho } d\mathrm{\phi } d\mathrm{\theta }}{{\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{2\sin \left(\mathrm{\phi }\right)}{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right) d\mathrm{\ρ} d\mathrm{\φ} d\mathrm{\θ}}$

$\=\frac{\frac{1280}{231}\mathrm{EllipticK}\left(\frac{1}{\sqrt{2}}\right)\mathrm{\π}}{2 {\mathrm{\π}}^2}$$$

$\=\frac{640}{231 \mathrm{\pi }}\mathrm{EllipticK}\left(\frac{1}{\sqrt{2}}\right)$

$\doteq 1.6352$

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Average Value≻Spherical

Table 8.2.12(a)   Solution by task template implementing the FunctionAverage command

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

Table 8.2.12(b)   Integration of $f$ over $R$ by visualization task template

$\left(\frac{1280}{231}\mathrm{EllipticK}\left(\frac{1}{2}\sqrt{2}\right)\mathrm{\π}\right)\/\left(2 {\mathrm{\π}}^2\right)$ = $\frac{640\mathrm{EllipticK}\left(\frac{\sqrt{2}}{2}\right)}{231\mathrm{\pi }}$$\stackrel{\text{at 5 digits}}{\to }$$1.6352$$$

Table 8.2.12(c)   Completion of the solution from first principles

Initialize

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

$f≔{\mathrm{\ρ}}^{3\/2}\:$

Apply the FunctionAverage command from the Student MultivariateCalculus package

$\mathrm{FunctionAverage}\left(f\,\mathrm{\rho }\=0..2 \sin \left(\mathrm{\phi }\right)\,\mathrm{\phi }\=0..\mathrm{\pi }\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)$

$\frac{640\mathrm{EllipticK}\left(\frac{\sqrt{2}}{2}\right)}{231\mathrm{\pi }}$

Use the MultiInt command to obtain $Q$, the integral of $f$ over $R$ 

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

$Q≔\frac{1280\mathrm{EllipticK}\left(\frac{\sqrt{2}}{2}\right)\mathrm{\pi }}{231}$

Use the MultiInt command to obtain $V$, the volume of $R$ 

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

$V≔2{\mathrm{\pi }}^2$

Divide $Q$ by $V$ and obtain a floating=point approximation via the evalf command

$Q\/V$ = $\frac{640\mathrm{EllipticK}\left(\frac{\sqrt{2}}{2}\right)}{231\mathrm{\pi }}$$$

$\mathrm{evalf}\left(Q\/V\right)$ = $1.635103860$$$