Answer
$21$
Work Step by Step
$E = (x,y,z) | 0 \leq x \leq 2, 0 \leq y \leq\ 1, 0 \leq z \leq 3$
Integrating with respect to $x$, then $y$, and then $z$
$\int \int \int (xy+z^{2}) dV = \int^{3}_{0} \int^{1}_{0} \int^{2}_{0} (xy+z^{2}) dx dy dz$
$= \int^{3}_{0} \int^{1}_{0} [\frac{x^{2}y}{2}+z^{2}x]^{x=2}_{x=0}dy dz$
$=\int^{3}_{0} \int^{1}_{0} 2y+2z^{2}dydz$
$=\int^{3}_{0} [y^{2} + 2z^{2}y]^{1}_{0}dz$
$=\int^{3}_{0} 1+2z^{2}dz$
$ =[z+\frac{2z^{3}}{3}]^{3}_{0} = 3+18+21$
