Let $M$ be a smooth oriented manifold of dimension $n$. Let $\alpha = \alpha_{[0]} + \alpha_{[1]} + ...+ \alpha_{[n]}$ be smooth differential form on $M$ such that $\alpha_{[I]} \in \mathcal{A}^i(M)$. By definition, we have $$\int_M \alpha := \int_M \alpha_{[n]}.$$
Suppose that $N$ is submanifold of $M$ of dimension $k$. How is the integral of $\alpha$ on $N$ defined, is it $\int_N \alpha = \int_N \alpha_{[k]}$ ?