Answer
$\dfrac{312,500 \pi}{7}$
Work Step by Step
Set up the integral.
$I=\int_0^{\pi} \int_{0}^{2 \pi}\int_{0}^{5} (\rho^4) (\rho^2) \sin \phi d\rho d\theta d\phi=\int_0^{\pi} \int_{0}^{2 \pi}\int_{0}^{5} (\rho^6) \sin \phi d\rho d\theta d\phi$
$=[\int_0^{\pi} \sin \phi d\phi][ \int_{0}^{2 \pi}d\theta][ \int_{0}^{5} \rho^6 d\rho] $
$=[-\cos \phi]_0^{\pi} [\theta]_{0}^{2 \pi} [\dfrac{\rho^7}{7}]_0^5$
$=[-\cos \pi+\cos 0] (1+1) (\dfrac{78125}{7})$
$=(4 \pi) (\dfrac{78125}{7})$
Thus, we have $I=\dfrac{312,500 \pi}{7}$
