Integral of a Differential Form over an Oriented k-Sub-Rectangle in Euclidean Space

Let $n,k\in\mathbb{N}$ with $k\le n$, let $(S,\varepsilon)$ be an 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 continuous differential $k$-form on $U$. Choose data as in Oriented k-Sub-Rectangle of Euclidean Space, so that $S=\lambda_R(R)$ for a standard $k$-rectangle

$$R=[a_1,b_1]\times\cdots\times[a_k,b_k]$$

and active coordinate indices $1\le i_1<\cdots<i_k\le n$. Write $\omega$ in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets. Let $f:U\to\mathbb{R}$ be the coefficient function of the basis form

$$dx_{i_1}\wedge\cdots\wedge dx_{i_k}.$$

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

$$F_1(t_2,\dots,t_k)=\int_{a_1}^{b_1} f(\lambda_R(t_1,t_2,\dots,t_k))\,dt_1,$$

where this is the one-dimensional Riemann integral on $[a_1,b_1]$; it exists because the integrand is continuous on $[a_1,b_1]$, hence 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

$$F_r(t_{r+1},\dots,t_k)=\int_{a_r}^{b_r} F_{r-1}(t_r,\dots,t_k)\,dt_r,$$

again in the one-dimensional Riemann sense of The Riemann Integral on a Closed Interval via Lower and Upper Integrals, whenever the integrand is viewed as a function of $t_r$ alone. The integral of $\omega$ over $(S,\varepsilon)$ is defined by

$$\int_{(S,\varepsilon)} \omega=\varepsilon F_k.$$

Equivalently,

$$\int_{(S,\varepsilon)} \omega = \varepsilon \int_{a_k}^{b_k}\cdots\int_{a_1}^{b_1} f(\lambda_R(t_1,\dots,t_k))\, dt_1\cdots dt_k,$$

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