Answer
$$\int_{-1}^{1}\int_{-1}^{1} \int_{-1}^{1} x^{2} y^{2} z^{2} d x d y d z =\frac{8}{27}$$
Work Step by Step
\begin{aligned} I&= \int_{-1}^{1}\int_{-1}^{1} \int_{-1}^{1} x^{2} y^{2} z^{2} d x d y d z \\
&=\frac{1}{3} \int_{-1}^{1} \int_{-1}^{1}\left[x^{3} y^{2} z^{2}\right]_{-1}^{1} d y d z \\
&=\frac{1}{3} \int_{-1}^{1} \int_{-1}^{1}\left[(2) y^{2} z^{2}\right]_{-1}^{1} d y d z \\
&=\frac{2}{3} \int_{-1}^{1} \int_{-1}^{1} y^{2} z^{2} d y d z\\
&=\frac{2}{9} \int_{-1}^{1}\left[y^{3} z^{2}\right]_{-1}^{1} d z\\
&=\frac{2}{9} \int_{-1}^{1}\left[(2) z^{2}\right]_{-1}^{1} d z\\
&=\frac{4}{9} \int_{-1}^{1} z^{2} d z\\
&=\left[\frac{4}{27} z^{3}\right]_{-1}^{1}\\
&=\frac{4}{27} (1-(-1))\\
&=\frac{8}{27} \end{aligned}
