# Change of Coordinates for Surface Area Integral?

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.

• Exactly where? The final example (number 4) is a cylinder, so no spherical coordinates there... – Hans Lundmark Sep 26 '12 at 6:50

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 $d\phi d\theta$.