How Does the Volume Element Change in Spherical Coordinates?

[fishingspree2]

Say I have a solid given using polar coordinates
I want to compute its volume.

We know that when switching from cartesian to polar, dV becomes \rho^{2}\sin\phi d\rho d\theta d\phi

But I am not converting from cartesian to polar, I am already in polar coordinates.

do I still have to carry the \rho^{2}\sin\phi that comes from the dV conversion formula?

 

[lugita15]

Yes, you do. That formula for dV is the volume element in spherical coordinates, whether you're converting into spherical coordinates or staying in spherical coordinates.