Answer
$\dfrac{162\pi}{5}$
Work Step by Step
$I=\int_0^{\pi} \int_{0}^{\pi}\int_{0}^{3} (\rho^2 \sin^2 \phi \sin^2 \theta) (\rho^2 \sin \phi) d\rho d\theta d\phi$
or, $I=\int_0^{\pi} \int_{0}^{\pi} [\dfrac{\rho^5}{5}\sin^3 \phi \sin^2 \theta]_0^3 d\theta d\phi$
or, $I=\int_0^{\pi} \int_0^{\pi} \dfrac{243}{5}( \sin^3 \phi \sin^2 \theta) d\theta d\phi$
or, $I=\int_0^{\pi} \int_0^{\pi} [ \dfrac{243}{5} \sin^3 \phi (\dfrac{1}{2}-\dfrac{1}{2} \cos 2 \theta) d\theta d\phi$
or, $I=\int_0^{\pi} \dfrac{243}{10} \pi \sin^3 \phi d\phi$
or, $=(\dfrac{243 \pi}{10} )\int_0^{\pi} \sin \phi -\sin \phi\cos^2 \phi$
$=\dfrac{162\pi}{5}$
