Solution
= Solution
Split the integrand into its three coordinate contributions. Each integrates independently over the rectangular box:
$$
\begin{aligned}
I&=\left(\frac23\right)(4)(6)+(2)\left(\frac{16}{3}\right)(6)
+(2)(4)(18)\\
&=16+64+144=\boxed{224}.
\end{aligned}
$$
For example, the last factor is $\int_{-3}^3z^2dz=18$. No spherical or cylindrical Jacobian is needed for this Cartesian <integral>.