  We introduce an integral representation of the Monge-Amp\`ere equation, which
leads to a new finite difference method based upon numerical quadrature. The
resulting scheme is monotone and fits immediately into existing convergence
proofs for the Monge-Amp\`ere equation with either Dirichlet or optimal
transport boundary conditions. The use of higher-order quadrature schemes
allows for substantial reduction in the component of the error that depends on
the angular resolution of the finite difference stencil. This, in turn, allows
for significant improvements in both stencil width and formal truncation error.
The resulting schemes can achieve a formal accuracy that is arbitrarily close
to $\mathcal{O}(h^2)$, which is the optimal consistency order for monotone
approximations of second order operators. We present three different
implementations of this method. The first two exploit the spectral accuracy of
the trapezoid rule on uniform angular discretizations to allow for computation
on a nearest-neighbors finite difference stencil over a large range of grid
refinements. The third uses higher-order quadrature to produce superlinear
convergence while simultaneously utilizing narrower stencils than other
monotone methods. Computational results are presented in two dimensions for
problems of various regularity.
