As I read this definition, I was wondering what it would mean if we were integrating a 0-form (a scalar function). According to this definition, it seem that if $A$ is open in $\mathbb R^k$, then we need $\omega$ to be a $k$-form. So are we even able to integrate 0-forms over $k$-dimensional manifolds, using this definition? I know that that Munkres defines the integral of a scalar function over a $k$-dimensional parametrised-manifold separately - prior to discussing manifolds - but I would think that this definition is a generalisation of the scalar variant, and should therefore match up in the case we integrate a 0-form.
I understand that only a $k$-form will actually yield a regular integral if $A\subset\mathbb R^k$, but I don't see how we would integrate an $r$-form over a $k$-dimensional manifold, when $r\neq k$ (if that's even possible at all). Could someone clarify?