Bachman's line integral versus classical line integral

[Rasalhague]

Bachman's "line integral" versus "classical line integral"

David Bachman A Geometric Approach to Differential Forms
http://arxiv.org/abs/math/0306194

When Bachman talks, in Appendix A, about "classical" line, surface, volume integrals, does he mean integrals of differential 0-forms (scalar fields) over 1-, 2- and 3-dimensional domains of R³. Is this the distinction he's making between the integrals of Appendix A (for which nonlinear differential forms are required) and the kind of line, surface, volume integrals he discussed in Chapter 5 (the kind to which Stokes' theorem applies), in which the integrand was a differential form of the same dimension as the domain of integration (for which linear differential forms suffice)?

 

[Rasalhague]

Bachman:

"The thing that makes (linear) differential forms so useful is the Generalized Stokes theorem. We don't have anything like this for non-linear forms, but that's not to say that they don't have their uses. For example, there is no differential 2-form on R³ that one can integrate over arbitrary surfaces to find their surface area."

In this thread, following quasar987's suggestion, I integrated a linear differential 2-form on R³ over a surface to find its area, didn't I? What is special about this surface that allowed me to find its area by integrating a linear differential 2-form, and why does Bachman use the same example, a 2-sphere embedded in R³, to demonstrate the need to integrate a nonlinear differential form if one wants to calculate surface area?!

Lee (Introduction to Smooth Manifolds):

Although differential forms are natural objects to integrate on manifolds, and are essential for use in Stokes' theorem, they have the disadvantage of requiring orientable manifolds.

The 2-sphere in R³ is orientable, isn't it? Likewise (the image of) a regular curve, and yet Bachman writes

Finally, we can define what is classically called a line integral as follows

$$\oint_C f(x,y) \; ds = \int_C f(x,y) \sqrt{dx^2+dy^2}$$

(The circle an accident? He doesn't mention anything about loops in particular.) Both Bachman and Lee talk about line integrals as one of the motivating applications of linear differential 1-forms, but in these his concluding words, Bachman seems to be going back on that and redefining a line integral as a nonlinear differential form. A whole methodology was developed for pulling back k-forms from a k-dimensional domain in Rⁿ to R^k where they can be treated as ordinary iterated integrals, a methodology which I thought was supposed to streamline and generalise "classical" line, surface, volume integrals; yet here, he seems to be saying that these methods can't be used for the very thing (line, surface, volume integrals) I thought they were designed for.

What is going on here?