So I have to compute the triple integral of this: $\int\int\int \frac{1}{1+x^2+y^2+z^2}$ and it says the equation of the sphere is $ x^2 + y^2 + z^2 = z$ which is just an elongated sphere running along the z-axis. But I don't understand how to setup the triple integral, the z on the ride side is totally throwing me off.
1) What would the bounds for $\rho$ be?
2) Why are the bounds for $\phi$ always 0 to $\frac{\pi}{2}$?
Any help would be appreciated, thanks.
This is not an elongated sphere, but just displaced so that it sits atop the plane $z=0$. The equation of the sphere in spherical coordinates is
$$\rho^2=z = \rho \cos{\phi} \implies \rho=\cos{\phi}$$
where $\phi \in [0,\pi/2]$ because the sphere is entirely in the half-space $z \ge 0$. The triple integral then takes the form
$$\int_0^{\pi/2} d\phi \, \sin{\phi} \: \int_0^{\cos{\phi}} d\rho \frac{\rho^2}{1+\rho^2} \: \int_0^{2 \pi} d\theta$$