A general method is presented for the evaluation of phase integrals of the Volterra model of stable predator-prey interactions for any fixed number N >= 2 of interacting species. The method is based on a bijective transformation of trajectory surfaces onto a flat hyperplane in EuclideanN-space, parametrized by a new set of dynamical variables in terms of which integration over the surface is straightforward. Expressions are displayed for the surface area of the trajectory surface and for the volume contained within this surface, quantities which are known to play an important role in the description of the statistical properties of Volterra trajectories.