Mass of body limited by spheres I need to find the mass of $E$, when $E$ is limited by $x^2+y^2+z^2 = 4z$ and $x^2+y^2+z^2 = 2z$ and it's density is $\rho(x,y,z) = (x^2+y^2+z^2)^{3/2}$.
Can anybody help me with the calculations. I don't even know how to get the limits of the integrals?

Transforming to spherical coordinates $(r,\theta,\phi)$ we see that the region is bounded by 
$$2\cos(\theta)\le r\le 4\cos(\theta)$$
with $0\le \theta\le \pi$ and $0\le \phi\le 2\pi$.
In addition, we see that $\rho(r,\theta,\phi)=r^3$.  Proceeding to find the mass, we write
$$\begin{align}
M_E&=\int_V \rho\,dV\\\\
&=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{2\cos(\theta)}^{4\cos(\theta)} \,r^3\left(\,r^2\sin(\theta)\right)\,dr\,d\theta\,d\phi\\\\
&=\frac{\pi}{3} (4^6-2^6)\int_0^\pi \cos^6(\theta)\,d\theta 
\end{align}$$
And you can carry out the last integral.
