Inspired by the introduction of Volumetric Modeling via volumetric representations (V-reps) by Massarwi and Elber in 2016, in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries, seen as a special class of more general V-reps. We develop tools for approximation and local re-parametrization of trimmed elements for three dimensional problems, and we provide a theoretical framework that fully justify our algorithmic choices. We validate our approach both on two and three dimensional problems, for diffuusion and linear elasticity.