Answer
$$6$$
Work Step by Step
Given $$ f(x, y, z)=z^{4} ; \quad 2 \leq x \leq 8, \quad 0 \leq y \leq 5, \quad 0 \leq z \leq 1$$
Then
\begin{aligned}
\iiint_{\mathcal{B}}f(x,y,z)dV&= \iiint_{\mathcal{B}} z^{4} d V \\
&=\int_{2}^{8} \int_{0}^{5} \int_{0}^{1} z^{4} d z d y d x\\
&=\int_{2}^{8} \int_{0}^{5}\left(\int_{0}^{1} z^{4} d z\right) d y d x\\
&=\left.\int_{2}^{8} \int_{0}^{5} \frac{1}{5} z^{5}\right|_{z=0} ^{1} d y d x \\
&=\int_{2}^{8} \int_{0}^{5} \frac{1}{5} d y d x\\
&=\frac{1}{5} \int_{2}^{8} \int_{0}^{5} d y d x\\
&=\frac{1}{5} \int_{2}^{8} y\bigg|_{0}^{5} d x\\
&=\int_{2}^{8} dx\\
&=x\bigg|_{2}^{8} \\
&= 6
\end{aligned}
