Answer
$$\frac{{52}}{{45}}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\int_0^{2{{\cos }^2}\theta } {\int_0^{4 - {r^2}} {r\sin \theta } dz} dr} d\theta \cr
& \int_0^{\pi /2} {\int_0^{2{{\cos }^2}\theta } {\left[ {\int_0^{4 - {r^2}} {r\sin \theta } dz} \right]} dr} d\theta \cr
& {\text{Integrate with respect to }}z \cr
& \int_0^{4 - {r^2}} {r\sin \theta } dz = r\sin \theta \left[ z \right]_0^{4 - {r^2}} \cr
& = r\sin \theta \left[ {4 - {r^2}} \right] \cr
& = 4r\sin \theta - {r^3}\sin \theta \cr
& \int_0^{\pi /2} {\int_0^{2{{\cos }^2}\theta } {\left[ {\int_0^{4 - {r^2}} {r\sin \theta } dz} \right]} dr} d\theta \cr
& = \int_0^{\pi /2} {\int_0^{2{{\cos }^2}\theta } {\left( {4r\sin \theta - {r^3}\sin \theta } \right)} dr} d\theta \cr
& {\text{Integrate with respect to }}r \cr
& = \int_0^{\pi /2} {\left[ {2{r^2}\sin \theta - \frac{1}{4}{r^4}\sin \theta } \right]_0^{2{{\cos }^2}\theta }} d\theta \cr
& = \int_0^{\pi /2} {\left[ {2{{\left( {2{{\cos }^2}\theta } \right)}^2}\sin \theta - \frac{1}{4}{{\left( {2{{\cos }^2}\theta } \right)}^4}\sin \theta } \right]} d\theta \cr
& = \int_0^{\pi /2} {\left( {8{{\cos }^4}\theta \sin \theta - 4{{\cos }^8}\theta \sin \theta } \right)} d\theta \cr
& {\text{Integrate}} \cr
& = \left[ { - \frac{{8{{\cos }^4}\theta }}{5} + \frac{{4{{\cos }^9}\theta }}{9}} \right]_0^{\pi /2} \cr
& = \left[ { - \frac{{8{{\cos }^4}\left( {\pi /2} \right)}}{5} + \frac{{4{{\cos }^9}\left( {\pi /2} \right)}}{9}} \right] - \left[ { - \frac{{8{{\cos }^4}\left( 0 \right)}}{5} + \frac{{4{{\cos }^9}\left( 0 \right)}}{9}} \right] \cr
& = \left[ 0 \right] - \left[ { - \frac{8}{5} + \frac{4}{9}} \right] \cr
& = \frac{{52}}{{45}} \cr} $$
