Answer
$$12\sqrt 3 $$
Work Step by Step
$$\eqalign{
& \int_0^3 {\int_0^{\pi /3} {\int_0^4 {r\cos \theta } dr} d\theta } dz \cr
& = \int_0^3 {\int_0^{\pi /3} {\left[ {\int_0^4 {r\cos \theta } dr} \right]} d\theta } dz \cr
& {\text{Integrate with respect to }}r \cr
& \int_0^4 {r\cos \theta } dr = \cos \theta \left[ {\frac{1}{2}{r^2}} \right]_0^4 \cr
& = \cos \theta \left[ {\frac{1}{2}{{\left( 4 \right)}^2}} \right] \cr
& = 8\cos \theta \cr
& \int_0^3 {\int_0^{\pi /3} {\left[ {\int_0^4 {r\cos \theta } dr} \right]} d\theta } dz = \int_0^3 {\int_0^{\pi /3} {8\cos \theta } d\theta } dz \cr
& {\text{Integrate with respect to }}\theta \cr
& \int_0^3 {\int_0^{\pi /3} {8\cos \theta } d\theta } dz = \int_0^3 {\left[ {8\sin \theta } \right]_0^{\pi /3}} dz \cr
& = \int_0^3 {\left[ {8\sin \left( {\frac{\pi }{3}} \right) - 8\sin \left( 0 \right)} \right]} dz \cr
& = \int_0^3 {\left[ {8\sin \left( {\frac{{\sqrt 3 }}{2}} \right)} \right]} dz \cr
& = 4\sqrt 3 \int_0^3 {dz} \cr
& {\text{Integrate}} \cr
& {\text{ = }}4\sqrt 3 \left[ {3 - 0} \right] \cr
& {\text{ = }}12\sqrt 3 \cr} $$
