We introduce a new way of approximating initial condition to the semidiscrete finite element method for integro-differential equations using any degree of elements. We obtain several superconvergence results for the error between the approximate solution and the Ritz-Volterra projection of the exact solution. For k > 1, we obtain first order gain in Lp(2≤ p ≤∞) norm, second order in W1,p(2≤ p ≤...