I have seen in several answers to questions on this page stating that there is no way to integrate a $k$-form over an oriented smooth $n$-manifold if $k \neq n$. However I cite Tu in his book on manifolds (page 272):
(Problem 23.3*) Suppose $N$ and $M$ are connected, oriented $n$-manifolds and $F : N \to M$ is a diffeomorphism. Prove that for any $\omega \in \Omega^k_c(M)$, $$ \int_N F^* \omega = \pm \int_M \omega $$ where the sign depends on whether $F$ is orientation-preserving or orientation-reversing.
That means it seems there is a way to make this definition in the case that $M,N$ is connected. But, does Tu mean that $n = k$? If not, is there some canonical way to make the definition such that can understand what he is asking to prove?