Give an explicit formula for $\mathcal{J}$ which makes the following result hold:

$$\int_{D} f d x d y d z=\int_{D^{\prime}} \phi|\mathcal{J}| d u d v d w$$

where the region $D$, with coordinates $x, y, z$, and the region $D^{\prime}$, with coordinates $u, v, w$, are in one-to-one correspondence, and

$$\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))$$

Explain, in outline, why this result holds.

Let $D$ be the region in $\mathbb{R}^{3}$ defined by $4 \leqslant x^{2}+y^{2}+z^{2} \leqslant 9$ and $z \geqslant 0$. Sketch the region and employ a suitable transformation to evaluate the integral

$$\int_{D}\left(x^{2}+y^{2}\right) d x d y d z$$