Finding the mass of a body with changing density

Question

How do I find the mass of a body that is outside the sphere $x^2+y^2+z^2=2z$, but inside $x^2+y^2+z^2=4, z ≥ 0$. With the density $ρ(x,y,z) = z\sqrt{x^2+y^2+z^2}$

Answer 1

If $V$ is the given region then mass of an object with density $\rho$ is given by,

$$\text{Mass}=\iiint_V\rho dV$$

For the given problem its advisable to convert to spherical coordinates and proceed. I hope you can at least do the computations.

Answer 2

Use cylindrical coordinates $(r,\phi, z)$ with $r:=\sqrt{x^2+y^2}$. Then $${\rm d}M=\rho(r,\phi,z){\rm d}V=z\sqrt{r^2+z^2}\> 2\pi\>r\>dr\>dz\ ,$$ since we have rotational symmetry with respect to the $z$-axis. Horizontal planes $z={\rm const.}$ intersect the solid in question in circular annuli. We therefore obtain $$M=2\pi\int_0^2 z\>\int_{\sqrt{2z-z^2}}^{\sqrt{4-z^2}} \sqrt{r^2+z^2}\>r\>dr\>dz\ .$$