A high-order method of moment solution with quadrature-point-based sampling is presented for the solution of the volume electric field integral equation for the scattering of inhomogeneous dielectric bodies. The proposed scheme efficiently allows for the material profile to be inhomogeneous within a curvilinear cell. It is demonstrated that the method leads to exponential convergence in both the radar cross section (RCS) and the volume current density. It is also demonstrated that the method can be more efficient than surface field integral equation formulations for thin-material scattering.