  In this paper, both semidiscrete and completely discrete finite volume
element methods (FVEMs) are analyzed for approximating solutions of a class of
linear hyperbolic integro- differential equations in a two-dimensional convex
polygonal domain. The effect of numerical quadrature is also examined. In the
semidiscrete case, optimal error estimates in L^{\infty}(L2) and
L^{\infty}(H1)- norms are shown to hold with minimal regularity assumptions on
the initial data, whereas quasi-optimal estimate in derived in
L^{\infty}(L^{\infty})-norm under higher regularity on the data. Based on a
second order explicit method in time, a completely discrete scheme is examined
and optimal error estimates are established with a mild condition on the space
and time discretizing parameters. Finally, some numerical experiments are
conducted which confirm the theoretical order of convergence.
