Change of Coordinates for Surface Area Integral?
If you are referring to Example 2 (the only example with spherical), then notice that he does conclude the surface element changes is $r^2\sin(\phi) d\phi d\theta$, where he calculates that $\|r_{\theta} \times r_{\phi}\| = 4 \sin{\phi}$. This is precisely what you want, since the radius of the sphere was $r = 2$. In that problem, he uses $dA$ to mean $d\phi d\theta$.