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A solid lies inside the cylinder $r=2$, within the sphere $$x^{2}+y^{2}+z^{2}=16,$$ and above the xy-plane. The density at a point $P$ is directly proportional to the distance from the xy-plane. Find the mass and the moment of inertia $I_z$.

How do I go about this? I'm not sure if I should use spherical or cylindrical coordinates.

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  • $\begingroup$ Start with a sketch of the volume and its boundaries. Notice that the boundaries of the volume are easy to describe in cylindrical coordinates, but a little more difficult to describe in spherical coordinates. Do the integration along the z-direction first. $\endgroup$
    – LouisB
    Dec 5 '15 at 22:07
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You shall plit the integral in two integrals. Your solid is a cylinder with a cap. You shall integrate the function(for the inertia moment) $\rho r^2=z(x^2+y^2+z^2)$ on these two regions:

$V_1$: the cylinder , with radius $2$ , with $z\in [0,3\sqrt{2}]$;

$V_2$(the cap) : $z\in [3\sqrt{2},\sqrt{(16-x^2-y^2)}]$ .

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  • $\begingroup$ shouldn't the function to integrate for the moment of inertia be $(x^2+y^2)(\sqrt{x^2+y^2+z^2})$ $\endgroup$
    – Upstart
    Jul 30 '17 at 19:38

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