Let $M$ be a compact orientable manifold of dimension $n$. Then, there is a canonical orientation class $$or_M \in H^n(M,\mathbb{Z})$$ in singular homology. Then we have $$\int_M \omega = \int_{or_M} \omega,$$ for any differential top form.
Now assume that $M$ is orientable but not compact and that we have a compact subset $K \subset M.$ There is a canonical orientation class $$or_{M,K} \in H^n(M,M \backslash K, \mathbb{Z})$$ in relative singular homology. Now take a top differential form which vanishes on $M \backslash K$. (Hence integrable on $M$.) We have $$\int_{M} \omega = \int_{or_{M,K}} \omega.$$ How to prove this last equality ? I don't even know what is the meaning of the right hand side. Thanks for any help.