On the integration of differential forms In my notes the integration of a differential form on an oriented manifold $M$, with  $\{(U_\alpha,\phi_\alpha) \}$ oriented atlas, is defined as: $\int_M \omega = \sum_{i \in \mathbb{N}}\int_M \omega_i$, where $\omega_i=\rho_i \omega$, where $\rho$ is a partition of unity subordinate to the cover $\{U_\alpha \}$. (For every $i$ there is $\alpha(i)$ s.t. supp $\rho_i$ is in $U_{\alpha (i)}$).
But why this sum is finite? I know that the set of supp $\omega_i$ is locally finite and $\omega$ has compact support but I think that I have however to sum on all $i$... Someone may help me?

Finite sum $\ne$ finite number of summands. If $M$ is compact, you have a finite cover, so a finite number of summands. And consider the 0-form (function) $\omega(x)=1\ \forall x\in M={\Bbb R}$.
EDIT:
The partition of unity is required to define the integral. In each "patch":
$$
\int_{patch}\omega_i =\int_{{\rm subset of } R^n}\omega_i{\rm -composed-with-bijection}
$$
See http://www.math.cornell.edu/~sjamaar/papers/manifold.pdf, definition 12.1.
