Answer
$$\frac{{27}}{2}$$
Work Step by Step
$$\eqalign{
& \int_0^3 {\int_{\pi /2}^\pi {\int_2^5 {z\sin x} dy} dx} dz \cr
& = \int_0^3 {\int_{\pi /2}^\pi {\left[ {\int_2^5 {z\sin x} dy} \right]} dx} dz \cr
& {\text{Integrate with respect to }}y \cr
& \int_2^5 {z\sin x} dy = \left[ {zy\sin x} \right]_2^5 \cr
& = 5z\sin x - 2z\sin x \cr
& = 3z\sin x \cr
& \int_0^3 {\int_{\pi /2}^\pi {\left[ {\int_2^5 {z\sin x} dy} \right]} dx} dz = \int_0^3 {\int_{\pi /2}^\pi {3z\sin x} dx} dz \cr
& {\text{Integrate with respect to }}x \cr
& \int_{\pi /2}^\pi {3z\sin x} dx = \left[ { - 3z\cos x} \right]_{\pi /2}^\pi \cr
& = - \left[ {3z\cos \left( \pi \right) - 3z\cos \left( {\frac{\pi }{2}} \right)} \right] \cr
& = - \left( { - 3z - 0} \right) \cr
& = 3z \cr
& \int_0^3 {\int_{\pi /2}^\pi {3z\sin x} dx} dz = \int_0^3 {3z} dz \cr
& {\text{Integrate}} \cr
& {\text{ = }}\left[ {\frac{3}{2}{z^2}} \right]_0^3 \cr
& = \frac{3}{2}{\left( 3 \right)^2} \cr
& = \frac{{27}}{2} \cr} $$
