  We examine the effect of numerical integration on the convergence of high
order pyramidal finite element methods. Rational functions are indispensable to
the construction of pyramidal interpolants so the conventional treatment of
numerical integration, which requires that the finite element approximation
space is piecewise polynomial, cannot be applied. We develop an analysis that
allows the finite element approximation space to include rational functions and
show that despite this complication, conventional rules of thumb can still be
used to select appropriate quadrature methods on pyramids. Along the way, we
present a new family of high order pyramidal finite elements for each of the
spaces of the de Rham complex.
