Let $n,k\in\mathbb{N}$ with $k\le n$, let $(S,\varepsilon)$ be an \reftext{def:oriented-k-sub-rectangle-euclidean-2026a}{oriented $k$-sub-rectangle} of $\mathbb{R}^n$, let $U\subseteq \mathbb{R}^n$ be open with $S\subseteq U$, and let $\omega$ be a \reftext{def:continuous-differential-k-form-euclidean-open-set-2026b}{continuous differential $k$-form} on $U$. Choose data as in \ref{def:oriented-k-sub-rectangle-euclidean-2026a}, so that $S=\lambda_R(R)$ for a standard $k$-rectangle

and active coordinate indices $1\le i_1<\cdots<i_k\le n$. Write $\omega$ in the coordinate expansion from \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b}. Let $f:U\to\mathbb{R}$ be the coefficient function of the basis form

For each fixed $(t_2,\dots,t_k)\in [a_2,b_2]\times\cdots\times[a_k,b_k]$, define

where this is the one-dimensional \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} on $[a_1,b_1]$; it exists because the integrand is continuous on $[a_1,b_1]$, hence \reftext{lem:continuous-implies-riemann-integrable-c54-2026b}{Riemann integrable}. Recursively, for each $r\in\{2,\dots,k\}$ and each fixed $(t_{r+1},\dots,t_k)\in [a_{r+1},b_{r+1}]\times\cdots\times[a_k,b_k]$, define

again in the one-dimensional Riemann sense of \ref{def:riemann-integrable-closed-interval-c54-2026b}, whenever the integrand is viewed as a function of $t_r$ alone. The integral of $\omega$ over $(S,\varepsilon)$ is defined by

Equivalently,

with the right-hand side understood through the recursive construction above.