Answer
$21$
Work Step by Step
Here, we have $E = (x,y,z) | 0 \leq x \leq 2, 0 \leq y \leq\ 1, 0 \leq z \leq 3$
Consider $I=\iiint (xy+z^{2}) dV $
Now integrate the integral first with respect to $x$, then $y$, and then $z$
$I= \int^{3}_{0} \int^{1}_{0} \int^{2}_{0} (xy+z^{2}) dx dy dz= \int^{3}_{0} \int^{1}_{0} [\dfrac{x^{2}y}{2}+z^{2}x]^{2}_{0}dy dz$
Now, we have
$I=\int^{3}_{0} \int^{1}_{0} 2y+2z^{2}dydz=\int^{3}_{0} [y^{2} + 2z^{2}y]^{1}_{0}dz$
This implies that $I=\int^{3}_{0} 1+2z^{2}dz=[z+\dfrac{2}{3}z^{3}]^{3}_{0} = 21$
