Answer
$$\frac{2}{3}{\pi ^2}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\int_0^{\pi /2} {\int_0^2 {{\rho ^2}} d\rho d\theta d\phi } } \cr
& \int_0^{\pi /2} {\int_0^{\pi /2} {\left[ {\int_0^2 {{\rho ^2}} d\rho } \right]d\theta d\phi } } \cr
& {\text{Integrate with respect to }}\rho \cr
& \int_0^2 {{\rho ^2}} d\rho = \left[ {\frac{1}{3}{\rho ^3}} \right]_0^2 \cr
& = \frac{1}{3}{\left( 2 \right)^3} - \frac{1}{3}{\left( 0 \right)^3} = \frac{8}{3} \cr
& = \int_0^{\pi /2} {\int_0^{\pi /2} {\frac{8}{3}d\theta d\phi } } \cr
& = \frac{8}{3}\int_0^{\pi /2} {\int_0^{\pi /2} {d\theta d\phi } } \cr
& {\text{Integrate with respect to }}\theta \cr
& = \frac{8}{3}\int_0^{\pi /2} {\left[ \theta \right]_0^{\pi /2}d\phi } \cr
& = \frac{8}{3}\int_0^{\pi /2} {\frac{\pi }{2}d\phi } \cr
& = \frac{4}{3}\pi \int_0^{\pi /2} {d\phi } \cr
& {\text{Integrate}} \cr
& {\text{ = }}\frac{4}{3}\pi \left[ {\frac{\pi }{2} - 0} \right] \cr
& {\text{ = }}\frac{2}{3}{\pi ^2} \cr} $$
