In this lecture we explain how to make sense of integrating a differential form on a manifold. We define $\int_M \omega$ only when:

• $M$ is oriented,
• $\omega$ is of the same degree as the dimension of $M$,
• $\omega$ is compactly supported.

We then state and prove the global version of Stokes’ Theorem.

Afterwards, we show that de Rham cohomology is a homotopy invariant, and use this to prove a very useful statement known as the Poincaré Lemma: every closed differential form is locally exact.