I teach mostly physics and a little math at a California community college. I've never taught the multivariate calculus course, but I have taught the electricity and magnetism course for which the math course is largely a service course. I think our curriculum is fairly standard for the US: after taking a year of single-variable calculus, students take a 3rd-semester course covering multivariable calculus.
In a typical math course of this type, what kind of area integrals do students actually learn to do?
In the E&M course, 90% of what we use area integrals for is just cases where we integrand is constant. We do a few simple examples that have some special symmetry, so that they can find the electric field of a given charge distribution. A typical case like this would be determining the field of a uniform line of charge, in which we pick the Gaussian surface to be a cylinder. Students mainly do minor variations on this, such as the field of a uniformly charged cylinder. There are maybe half a dozen simple examples of this type that you can make up, and that's about it.
At the opposite extreme, it seems like the general case of evaluating such a surface integral would involve basically treating the surface as a manifold (even if we don't use the word) and picking coordinate charts on it (even if we don't call it that). Most surfaces can't be covered with a single chart, so in general we'd have to use multiple charts.
Are students actually taught to do something in between these extremes of complexity? For example, does the typical math text restrict attention to cases in which the surface can be projected onto a coordinatized plane in such a way that the mapping is either 1-to-1 or 2-to-1 (as for a sphere, but not for most non-convex surfaces)?