Question

When using Stroke's theorem using spherical coordinates, does the integral end with [r^2]sin[phi] (before you write d[r]d[phi]d[theta])?

Answer 1

Did you mean Stokes Theorem?

In rectangular coordinates we use a differential cube with sides dx * dy * dz.
For any integral in spherical coordinates we need a similar cube. One side is of length dr. Another is r*d[phi]. The third is r*sin(phi)*d[theta].