TITLE: What is the theory of non-linear forms (as contrasted to the theory of differential forms)?
QUESTION [23 upvotes]: It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=\sqrt{dx^2+dy^2}$, the unsigned 1-d forms $|f(x,y)dx+g(x,y)dy|$, or the unsigned area forms $|h(x,y)dx\wedge dy|$. 
My question is:

Where do the arclength form $ds=\sqrt{dx^2+dy^2}$, the unsigned 1-d forms |f(x,y)dx+g(x,y)dy|, and the unsigned area forms $|h(x,y)dx\wedge dy|$ live relative to the differentials $dx$ and $dy$, which I understand to live in the cotangent bundle of some 2-dimensional manifold?

REPLY [2 votes]: In my opinion, you're looking for the notion of a cogerm.
If I understand correctly, the fact that such things act on paths (and not just vectors) allows for "higher order" forms like $d^2 x$, and the fact that such things aren't assumed linear allows for "non-linear" forms like $ds := \sqrt{dx^2+dy^2}$. And yes, there is indeed a notion of integration for such forms; see the link.