Answer
$$105$$
Work Step by Step
Given $$f(x, y, z)=x z^{2} ; \quad[-2,3] \times[1,3] \times[1,4]$$
Then
\begin{align*}
\iiint_{\mathrm{B}} f(x, y, z) d V&=\int_{-2}^{3} \int_{1}^{3} \int_{1}^{4} x z^{2} d z d y d x\\
&=\frac{1}{3} \int_{-2}^{3} \int_{1}^{3} x z^{3} \bigg|_{1}^{4} d y d x\\
&= 21\int_{-2}^{3} \int_{1}^{3} x d y d x\\
&= 21\int_{-2}^{3} x y\bigg|_{1}^{3} d x\\
&= 42\int_{-2}^{3} x d x\\
&=21 x^2\bigg|_{-2}^{3}\\
&= 105
\end{align*}
