Integrating a density over a Mobius strip

Question

According to this link one can integrate over a Mobius strip by using "densities". That has me very excited but I can't seem to find a reference on this. Can someone provide a book/ online source that explains in detail what densities are, how they relate to regular differential forms, and shows how one can integrate over non-orientable manifolds (with boundary) with them?

Alternatively, if it wouldn't take too much effort to explain it here, that'd also work. ;)

Thanks.

Answer 1

You might look at Ted Frankel's book, as well as Abraham-Marsden-Ratiu.