In [CD12], integration is described in terms of a contraction: Similarly as in differential geometry, we may view a superform $\alpha\in A^{p,q}(U)$ as a multilinear map

$$N_{\mathbb{R}}^{p+q}\longrightarrow C^{\infty}(U),\quad(n_{1},\dots,n_{p+q})\mapsto\alpha(n_{1},\dots,n_{p+q})$$

which is alternating in the variables $(n_{1},\dots,n_{p})$ and also in $(n_{p+1},\dots,n_{p+q})$. Let $I\subset\{1,\dots,p+q\}$ be a subset of cardinality $s$ with $s^{\prime}$ elements contained in $\{1,\dots,p\}$ and hence $s^{\prime\prime}=s-s^{\prime}$ elements in $\{p+1,\dots,p+q\}$. Given vectors $v_{1},\dots,v_{s}\in N_{\mathbb{R}}$, the contraction $\langle\alpha;v_{1},\dots,v_{s}\rangle_{I}\in A^{p-s^{\prime},q-s^{\prime\prime}}(U)$ is given by inserting $v_{1},\dots,v_{s}$ for the variables $(n_{i})_{i\in I}$ of the above multilinear function.

Using the basis $e_{1},\dots,e_{r}$ of $N$ and assuming $\alpha\in A_{c}^{r,r}(U)$, the contraction $\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{r+1,\dots,2r\}}$ is a $(r,0)$-superform which may be viewed as a classical $r$-form on $U$. Then it is immediately clear from the definitions that we have

$$\int_{U}\alpha=\int_{U}\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{r+1,\dots,2r\}}$$

where we use the usual integration of $r$-forms on the right. Of course, there is no preference to contract with respect to the last $r$ variables. Similarly, may view $\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{1,\dots,r\}}\in A_{c}^{0,r}(U)$ as a classical $r$-form and we have

$$\int_{U}\alpha=\int_{U}\langle\alpha;e_{1},\dots,e_{r}\rangle_{\{1,\dots,r\}}.$$