When learning vector fields and using Green's Theorem with the Jacobian to find the area of a level surface, I actually realized that most of the examples shown in my book would be much easier to solve by using polar and spherical coordinates, but not multiplying by the representation of the radius. (Eg: In spherical coordinates I would eliminate the $\rho^2 \sin(\phi)$. I ended up discussing with my professor about it, and he said that the radius should be fixed. I disagreed, saying that the radius is changing, but you are not multiplying by it.
Here is how I see the use of Spherical Coordinates for getting the volume:
My idea is that, in spherical coordinates, if you take $z = \rho \cos(\theta)$, and look at the Cartesian plane from the z-axis, you are actually adding all the circles on the xy-plane. Then by looking from the y-axis, you have other sets of circles, and that's where you are getting your radius for multiplying $\rho^2 \sin(\phi)$. Is this right or I understood it wrongly?
Thanks for your attention