  In this paper we analyze the approximation of multivariate integrals over the
Euclidean plane for functions which are analytic. We show explicit upper bounds
which attain the exponential rate of convergence. We use an infinite grid with
different mesh sizes and lengths in each direction to sample the function, and
then truncate it. In our analysis, the mesh sizes and the truncated domain are
chosen by optimally balancing the truncation error and the discretization
error.
  This paper derives results in comparable function space settings, extended to
$\R^s$, as which were recently obtained in the unit cube by Dick, Larcher,
Pillichshammer and Wo{\'z}niakowski (2011).
  They showed that both lattice rules and regular grids, with different mesh
sizes in each direction, attain exponential rates, hence motivating us to
analyze only cubature formula based on regular meshes.
  We further also amend the analysis of older publications, e.g., Sloan and
Osborn (1987) and Sugihara (1987), using lattice rules on $\R^s$ by taking the
truncation error into account and extending them to take the anisotropy of the
function space into account.
