What is Integral of a form?

Let $M$ be an oriented $k$-submanifold and $\omega$ a $k$-form defined on a neighborhood of $M$, with $\operatorname{supp}\omega \cap M$ compact. (a) If $\operatorname{supp}\omega \cap M \subseteq \gamma(V)$ for a single direct parametrization, set

— the right side being the Lebesgue integral over $V$ (Chapter 11) of the coefficient $g$ of $\gamma^*\omega = g\,\dd u_1\wedge\dots\wedge\dd u_k$, which is continuous with compact support. (b) In general, choose finitely many direct parametrizations $\gamma_i(V_i)$ covering the compact $\operatorname{supp}\omega \cap M$ and a subordinate partition of unity $(\chi_i)$ (Lemma 21.20), and set $\int_M\omega = \sum_i\int_M\chi_i\,\omega$, each term computed by (a). For a curve ($k = 1$) parametrized by $\gamma\colon\intcc ab\to\R^n$ we write $\int_\gamma\omega = \int_a^b\gamma^*\omega$, no injectivity required.