Integrate the function:
$$f(x,y,z)=1/\sqrt{(x^2+y^2)(x^2+y^2+z^2)}$$
over the region $R$ which is the set of all points outside the sphere $x^2+y^2+(z-1)^2=1$ but inside the sphere $x^2+y^2+z^2=4$.
I am pretty sure that spherical coordinates are the way to go because I realized that they make the intergrand equal to 0. However I am unsure of my limits of integration. I tried the following:
$\theta$ limits of integration: 0 to 2$\pi$,
$\phi$ limits of integration: 0 to $\pi$,
$\rho$ limits of integration: 2cos$\phi$ to 2,
My result was $4\pi^2$.