Functions on a manifold cannot be integrated per se. This is also the reason for the notation

${\displaystyle {\int }_{U}fd{x}_1\dots d{x}_n}$

if one wants to integrate a function $f$ over a region $U\subset {\mathbb{R}}^n$. If you express $f$ in different coordinates  (for example polar coordinates) the integral changes (unless you stick in some Jacobi determinant). Objects living on regions in ${\mathbb{R}}^n$ that can be integrated independent of the choice of coordinates are called $n$-forms. In the above formula $d{x}_1\dots d{x}_1$ is such an $n$-form and its integral over $U\subset {\mathbb{R}}^n$ always calculates the n-dimensional volume of $U$, no matter which coordinates you use. Conversely, if you know the integral of an n-form over any subregion of $U$ you know the $n$-form.

This gives us a clue how to define 2-forms on our discrete surfaces: A 2-form $\sigma$ assigns to each face $\phi$ a real number $\sigma (\phi )$. We also use the notation

${\displaystyle \sigma (\phi )=:{\int }_{\phi }\sigma }$.

If $U\subset F$ is a union of faces we define

${\displaystyle {\int }_U\sigma :=\sum _{\phi \in U}{\int }_{\phi }\sigma }$.

Also we write

${\displaystyle {\int }_M\sigma :={\displaystyle {\int }_F\sigma }}$.

In many situations it is possible to an area to each face. Such an assignment of areas to faces is a special case of a 2-form, usually referred to as the volume form ${\sigma }_{\text{vol}}$ of $M$. The total area of $M$ is then

${\displaystyle \text{vol}(M):={\int }_M{\sigma }_{\text{vol}}}$.

Even in the presence of a volume form discrete functions $f\in {\mathrm{\Omega }}_0(M)$ cannot be integrated. This is because they represent function values at vertices and there is no canonical way to multiply such an $f$ with ${\sigma }_{\text{vol}}$ in order to get a 2-form that can be intergrated.

We use this as a motivation to introduce another kind of objects: dual 0-forms. We think of them as real-valued functions on the surface that are constant on the interior of all faces. Formally speaking they are also just defined as functions

$f:F\to \mathbb{R}$.

The above remarks should have made it clear though that it is not wise to confuse dual 0-forms with 2-forms: functions cannot be integrated while this is perfectly possible for 2-forms. Functions can however be multiplied with 2-forms, the result being another 2-form:

${\displaystyle {\int }_{\phi }(f\sigma ):=f(\phi ){\int }_{\phi }\sigma }$.

Note that this seems to be a reasonable formula for functions $f$ that are constant on each face.

We denote the vector space of all functions by ${\mathrm{\Omega }}_0(M^{\ast })$. This space is indeed canonically isomorphic to the dual vector space of ${\mathrm{\Omega }}_1(M)$: To verify this statement it enough to exhibit a nondegenerate bilinear pairing

$⟨,⟩:{\mathrm{\Omega }}_0(M^{\ast })\times {\mathrm{\Omega }}_2(M)\to \mathbb{R}$

${\displaystyle ⟨f,\sigma ⟩:={\int }_{M}f\sigma }$.

Finally: If we are given a volume form, we can integrate dual functions by defining

${\displaystyle {\int }_{M}f:={\int }_{M}f{\sigma }_{\text{vol}}}$.

Integral of dual functions over unions $U$ of faces can then be defined in a similar way.