Let $M$ be a $n$ dimensional compact connected manifold. Let $\alpha$ be a differential form of degree $n$ such that $$ \int_{M} \alpha = 0 $$ then I would like to show that $\alpha$ vanishes at at least one point of $M$.
One could just say that a volume form can't give $0$ volume to the manifold but that's precisely the result whose proof I would like to understand.
I'm looking for an "elementary" answer if possible which doesn't use orientations (or if so explains the result used).
This question is linked to my previous question : Zeroes of exact differential forms on compact manifold indeed I am asking this question to understand the answer of the previous one.