Let's take the differential form $\omega = xy \, dx \wedge dy$. We say that $M$ is the surface $z = x^2 + y^2 \leq 1$ with the standard orientation.
I can calculate $\int_M \omega$ via pullback and get: $=\int_{0}^{2 \pi} \int_0^1 r^3 \sin(\theta) \cos(\theta) \,dr \,d\theta = 0.$
I can also try calculating it with the generalized Jacobian and got $$\int_0^{2 \pi} \int_0^1 r^3 \sin(\theta) \cos(\theta) \sqrt{1+4r^2} \, dr \, d\theta = 0.$$
Obviously the result is the same but only because of the domain. I think that there's a mistake in my thinking somewhere along the way, and that the results should be equal (on all domains).
Any help would be appreciated!