Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$}, let $\Omega$ be an admissible domain in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ with ambient set $D$ in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}, and let $\omega$ be a continuous differential $n$-form on $\Omega$ that is compactly supported in $\Omega$ in the sense of that definition.

By \ref{lem:zero-extension-box-integral-euclidean-2026a} there exists a \reftext{def:closed-box-rn-2026a}{closed box} $B\subseteq D$ containing $\operatorname{supp}\omega$, the iterated one-dimensional Riemann integrals of the zero extension $\tilde f$ of the coefficient function of $\omega$ over any such box exist, and the resulting value is independent of the chosen box. The \textbf{integral of $\omega$ over $\Omega$}, denoted

is defined to be this common value.