Answer
$\dfrac{8\pi}{3}+\dfrac{128}{15}$
Work Step by Step
Conversion of rectangular to cylindrical coordinate system gives: $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$
Here, $x=r \cos \theta; y=r \sin \theta, z=z$
Let $I=\iiint_E(x+y+z)dV=\int_0^{\pi/2} \int_{0}^{2}\int_{0}^{4-r^2} (x+y+z) r dr dz d\theta$
or, $I=\int_0^{\pi/2} \int_{0}^{2}\int_{0}^{4-r^2} (r \cos \theta+r \sin \theta+z) r dr dz d\theta$
or, $I=\int_0^{\pi/2} \int_{0}^{2}\int_{0}^{4-r^2}[r^2( \cos \theta+ \sin \theta+rz) dz dr d\theta=\int_0^{\pi/2} \int_{0}^{2}[r^2( \cos \theta+ \sin \theta)(z)+\dfrac{z^2 r}{2}]_{0}^{4-r^2} drd\theta$
or, $I=\int_0^{\pi/2}[\dfrac{64}{15}(\cos \theta+ \sin \theta)+\dfrac{16}{3} d\theta=\dfrac{8\pi}{3}+\dfrac{128}{15}$
