  Flat surfaces with erasing forest are obtained by deforming the flat metric
structure of translation surfaces, the moduli space of such surfaces is a
deformation of the moduli space of translation surfaces. On the moduli space of
flat surfaces with erasing forest, one can define some energy function
involving the area of the surface, and the total length of the erasing forest.
Note that on this space, we have a volume form which is defined by using
geodesic triangulations. The aim of this paper is to prove that the integral of
the energy functions mentionned above with respect to this volume form is
finite. As applications, we will use this result to recover some classical
results due to Masur-Veech, and Thurston.
