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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.

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    Try to change from cartesian to polar coordinates - this makes it a lot easier to describe the region $D$, as $\sqrt{x^2+y^2}\rightarrow r$. Find for which values of $r$ the inequality that describes $D$ can be satisfied - that is, for which values of $r$ is $r^2-16<4-r$. Then seperate the $z$-integral into two integrals - one for positive values of $z$ and one for negative values to get rid of the absolute value. Once you have done that, the integration itself should be fairly easy. – 2017-01-12
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    You mean a spherical coordinate system or a cylindrical one? – 2017-01-12
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    Cylindrical coordinates is what I meant, sorry for being unclear. Spherical just makes things complicated in the $z$ direction, which isn't necessary. – 2017-01-12
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    Thanks, I'll try this way then! – 2017-01-12
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    I got the result $-\dfrac{344}{3}\pi$... Hope it is the right one! – 2017-01-12

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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: $$ \int_D |z| dx dy dz =\iiint_{\Psi^{-1}(D)} |z|\rho d\rho d\theta dz=$$ $$=\int_0^{2\pi} \int_{-5}^4 \int _{\rho ^2-16}^{4-\rho} |z|\rho dz d\rho d\theta=$$ $$=2\pi \int_{-5}^4 \rho\int _{\rho ^2-16}^{4-\rho} |z| dz d\rho.$$

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    Thanks for the answer. Only, I don't understand why $\rho$ has to be $\in [4,5]$. Could you provide also an entire solution? I would be very grateful. Thx anyway! – 2017-01-12
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    @opisthofulax it's obvious I have mistaken. $\rho \in [-5,4]$ so when you integrate you must split the integral because of the absolute value of z. Sorry. I'm going to change the answer as soon as possible – 2017-01-13