Although it is a very simple question, but let me check my understand. The question is: Do we get zero if we integrate a differential form on a domain where the form vanishes?
Here is a simple example: Let us consider a differntial form $\omega$ in $\mathbb{R}^3$ and choose normal coordinates $(x,y,z)$: $$\omega \equiv dz \wedge dy - y^4 dx \wedge dy$$
The domain is the intersection $D:(z = x y^4) \cap (x^2 + y^2) \leq r^2$, for some positive constant $r$.
Hence everywhere on this domain $\omega =0$.
Then if we consider the integration of $\omega$ on $D$, I think I will get zero: $$\int_D \omega =0.$$
Is this correct? Actually, I'm worry about whether $\omega =0$ or not on the boundary $\partial D$.