Let $({\mathscr{C}},m)$ be a weighted integral ${\mathbb{R}}$-affine polyhedral complex of pure dimension $n$. For $\alpha\in A_{c}^{n,n}({\mathscr{C}})$, we set

$$\int_{({\mathscr{C}},m)}\alpha:=\sum_{\sigma\in{\mathscr{C}}_{n}}m_{\sigma}\int_{\sigma}\alpha,$$

where we use integration from 2.4 on the right. We define integrals over the boundary of ${\mathscr{C}}$ for a superform $\beta$ in $A_{c}^{n-1,n}({\mathscr{C}})$ or in $A_{c}^{n,n-1}({\mathscr{C}})$ by

$$\int_{\partial({\mathscr{C}},m)}\beta=\sum_{\sigma\in{\mathscr{C}}_{n}}m_{\sigma}\int_{\partial\sigma}\beta,$$

where we use the boundary integrals from 2.8 on the right. Note that the boundary $\partial{\mathscr{C}}$ may be defined as the subcomplex consisting of the polyhedra of dimension at most $n-1$, but there is no canonical weight on $\partial{\mathscr{C}}$. Indeed, the boundary integral $\int_{\partial({\mathscr{C}},m)}\beta$ depends on the relative situation $\partial{\mathscr{C}}\subset{\mathscr{C}}$ because of the weight $m_{\sigma}$ and the contraction with respect to the vectors $\omega_{\rho,\sigma}$ used in the definitions. This is similar to the situation in real analysis where boundary integrals depend on the relative orientation. These classical boundary integrals do depend only on the restriction of the differential form to the boundary which is clearly wrong for our boundary integrals. However, it is still true that $\int_{\partial({\mathscr{C}},m)}\beta=0$ if the support of $\beta$ is disjoint from $\partial{\mathscr{C}}$.