Evaluate $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$ where $W$ is the solid bounded by the two spheres $x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ where $0 < b < a$.
I can convert the integral to $\iiint_{W} \frac{\sin \phi d \phi d \theta d \rho}{\rho}$ but I don't know how to obtain the limits for this, and especially which limits match with which $d \phi, d \theta, d \rho,$ etc.
You convert to spherical coordinates, as you remark in the question. The limits for $\rho$ are easy it is from $a$ to $b$. $\phi$ ranges from the positive $z$ axis down to the negative $z$ axis so it is from $0$to $\pi$ And $\theta$ is a 360degrees on the x y plane so from $0$ to $2\pi$
Thus tou get $$\int_a^b \int_0^{\pi} \int_0^{2\pi} \frac{\sin \phi }{\rho} d \theta d\phi d\rho$$