This paper studies the asymptotic mean squared error for predicting the integral of a weakly stationary spatial process over a unit cube based on a centered systematic sample. For processes whose spectral density decays sufficiently slowly at infinity, the asymptotic mean squared error takes a form similar to that obtained by letting the cube increase in size with the number of observations. However, if the spectral density decays faster than a certain critical rate, then the asymptotic mean squared error takes on a completely different form. By adjusting the weights given to observations near an edge of the cube, it is possible to obtain asymptotic results for the fixed cube that again resemble those for the increasing cube.