For clarity of question, I'll pose an example.
Consider the region bounded by $z = \sqrt{x^2 + y^2}$ and $x^2 + y^2 + z^2 = 4$. Upon using spherical coordinates, my source says that $\rho$ should range from $0$ to $2$. I don't understand this, because when I draw the diagram, I see that the distance from origin to the surface enclosing this region is NOT always 2, as opposed to a sphere which would always have radius 2, and would allow $\rho$ to range from 0 to 2.
Why is $\rho$ from 0 to 2 and what would be the general method to finding the variation in $\rho$?
I think in 2-D and polar coordinates, the general method was to sub in $x = rcos(\theta)$ and $y = rsin(\theta)$ to find the relationship between $r$ and $\theta$. Is this similar for $\rho$?
Your first surface is a cone where for each value $z=k$, you have a circle centred at $0$ in this plane of radius $|k|$. Your second surface is a sphere of radius $2$ centred at the origin. The region bounded by these two surfaces looks like this in the region $z>0$ (to find it for $z<0$, just reflect it in the plane $z=0$):
If you travel from the origin outwards, you will never hit any part of the cone. You will eventually hit the surface of the sphere, by which point you will have travelled a distance of $2$. This is the case no matter what angle you travel at, as long as it is inside the cone. But this is dependent completely on the angle, and not on $\rho$, and so you don't need to worry about it for the range of $\rho$, which we can now conclude is $[0,2]$.