Answer
$\frac{27}{4}$
Work Step by Step
$B = (x,y,z) | 0 \leq x \leq 1, -1 \leq y \leq\ 2, 0 \leq z \leq 3$
Integrate with respect to $y$, then $z$, and then $x$
$\int \int \int xyz^{2} dV = \int^{1}_{0} \int^{3}_{0} \int^{2}_{-1} xyz^{2} dy dz dx$
$= \int^{1}_{0} \int^{3}_{0} [\frac{1}{2}xy^{2}z^{2}]^{y=2}_{y=-1}dz dx$
$=\int^{1}_{0} \int^{3}_{0} \frac{3}{2} xz^{2}dzdx$
$=\int^{1}_{0} [\frac{1}{2}xz^{3}]^{z=3}_{z=0}dx$
$=\int^{1}_{0} \frac{27}{2}xdx$
$ =[\frac{27}{4}x^{2}]^{1}_{0} = \frac{27}{4}$
