  Approximate solutions of linear and nonlinear integral equations using
methods related to an interpolatory projection involve many integrals which
need to be evaluated using a numerical quadrature formula. In this paper, we
consider discrete versions of the modified projection method and of the
iterated modified projection methodfor solution of a Urysohn integral equation
with a smooth kernel. For $r \geq 1,$ a space of piecewise polynomials of
degree less than or equal to r - 1 with respect to an uniform partition is
chosen to be the approximating space and the projection is chosen to be the
interpolatory projection at r Gauss points. The orders of convergence which we
obtain for these discrete versions indicate the choice of numerical quadrature
which preserves the orders of convergence. Numerical results are given for a
specific example.
