  We motive and calculate Newton--Cotes quadrature integration variance and
compare it directly with Monte Carlo (MC) integration variance. We find an
equivalence between deterministic quadrature sampling and random MC sampling by
noting that MC random sampling is statistically indistinguishable from a method
that uses deterministic sampling on a randomly shuffled (permuted) function. We
use this statistical equivalence to regularize the form of permissible Bayesian
quadrature integration priors such that they are guaranteed to be objectively
comparable with MC. This leads to the proof that simple quadrature methods have
expected variances that are less than or equal to their corresponding
theoretical MC integration variances. Separately, using Bayesian probability
theory, we find that the theoretical standard deviations of the unbiased errors
of simple Newton--Cotes composite quadrature integrations improve over their
worst case errors by an extra dimension independent factor $\propto N^{-1/2}$.
This dimension independent factor is validated in our simulations.
