Why do we need orientation to integrate on a manifold?

Question

In my differential geometry course we are studying the integration of compactly supported $n$-forms and I do not really get why we need our manifold to be oriented. Indeed, we define the integral of an $n$-form $\Omega$ as $$\int_M \Omega = \sum_i \int_{\varphi(U_i)} (\varphi_i^{-1})^*(\theta_i\Omega)$$ for $\{\theta_i\}$ some partition of unity subordinated to an open cover by domains of charts of an atlas compatible with the orientation. But it seems to me that we only need a local orientation on each domain of charts $U_i$ which is always possible on a manifold (actually, I'm not sure about that but intuitively we can define an orientation on each sufficiently small neighborhood of the Moebius band for example). What could possibly go wrong in this definition of the integral with a non-orientable manifold ?