I really can't figure out how to solve this triple integral $$ \iiint_D|z|dxdydz $$
on the domain $D$, that is so defined
$$ D = \left\{(x,y,z)\in\mathbb{R}^3: x^2+y^2-16 \leq z \leq 4 - \sqrt{x^2+y^2}\right\} $$
Any ideas or solutions? Thanks in advance.
By changing the coordinates to cylindrical ones with a diffeomorphism $\Psi $, $$ \Psi^{-1}(D)=\{(\rho,\theta,z): \rho ^2-16\leq z\leq 4-\rho\} = $$ $$ \{(\rho,\theta,z)\in \mathbb R_+ \times(0,2\pi)\times \mathbb R|\rho \in [-5,4], \ z \in [\rho ^2-16,4-\rho] \}. $$
$D$ is measurable and so is $\Psi^{-1}(D) $. So by changing coordinates and Fubini's theorem you have: $$ \int_D |z| dx dy dz =\iiint_{\Psi^{-1}(D)} |z|\rho d\rho d\theta dz=$$ $$=\int_0^{2\pi} \int_{-5}^4 \int _{\rho ^2-16}^{4-\rho} |z|\rho dz d\rho d\theta=$$ $$=2\pi \int_{-5}^4 \rho\int _{\rho ^2-16}^{4-\rho} |z| dz d\rho.$$