Question: 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)?

Answer: 
The following is based on my experience at as a TA and adjunct at a UC, a Cal State, and the University of Nevada, as well as observations of siblings and friends at other institutions.

The two texts which seem to be used most commonly are [Stewart's *Calculus*](https://books.google.com/books/about/Calculus_Early_Transcendentals.html?id=QyOYWR9RLI8C) and [Thomas' *Calculus*](https://books.google.com/books?id=yavIPgAACAAJ&dq=thomas+calculus) (note that both Stewart and Thomas have been dead for a while, and neither has been an author on either text in a decade or two—these are pretty well established franchises).

In my experience, both of these books give a somewhat cursory introduction to techniques of integration in multiple variables. They give a basic overview of the theory (in terms of Riemann sums over partitions of higher dimensional domains of integration), then quickly introduce Fubini / Tonelli so that students can actually compute some integrals.

The next several sections deal with the [generalized change of variables formula](https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables), with an emphasis on polar and spherical coordinates. The actual general change of variables formula is there, but is often elided in practice (as students often don't really have the necessary background in linear algebra to handle a more general change of coordinates). Finally, both texts include sections on vector fields, line and surface integrals, and the big theorems of multivariable integration (Stokes, Green, Gauss, etc). In my experience, these sections are typically taught very quickly near the end of the quarter / semester.

Because linear algebra is not typically a prerequisite for multivariable calculus, many of the results that you are looking for are likely to be beyond the ken of the average student. They likely won't know what a projection is, let alone a manifold or a cover by smooth charts. I believe that the goal is to give students a very brief overview of the general shape of calculus in higher dimensions, but that specific computations get left for future classes in, for example, physics departments.

Long story short: if it can't be done relatively simply in rectangular, polar, cylindrical, or spherical coordinates, or it cannot be done via an application of Stoke's theorem (and other related theorems) in a relatively quick way, then it is likely that the students haven't seen it.
How about learning to manipulate the formula
$$\int\_S\|\partial\_v\times\partial\_w\|dv\wedge dw$$
for students of vector calculus?
They, then, could be able calculate the area of the sphere, toroid, ellipsoid, ovoid in a well structured vector language, and volumes, with the aid of surface integral like
$$\iiint\_Rdx\wedge dy\wedge dz=\iint\_Sz\ dy\wedge dz$$
coming from the Divergence Theorem, for a bounded region $R$ with bounding surface $S$.

Observe that modern books are struggling to advance to the Cartan's Calculus thematics and techniques.