When using the formula for the surface area integral and using some change of coordinates, i.e. spherical coordinates, does the area element dA change to $r^2sin(\phi)\partial\phi\partial\theta$ ? Because I noticed in Dr. Paul Lamar's final example here he doesn't use the spherical area element. He simply uses the area element $\partial\phi\partial\theta$.
That's confusing to me. I feel like if he does a spherical change of coordinates, the area element he should be using is the spherical area element.