  This work presents a high-accuracy, mesh-free, generalized Stokes
theorem-based numerical quadrature scheme for integrating functions over
trimmed parametric surfaces and volumes. The algorithm relies on two
fundamental steps: (1) We iteratively reduce the dimensionality of integration
using the generalized Stokes theorem to line integrals over trimming curves,
and (2) we employ numerical antidifferentiation in the generalized Stokes
theorem using high-order quadrature rules. The scheme achieves exponential
convergence up to trimming curve approximation error and has applications to
computation of geometric moments, immersogeometric analysis, conservative field
transfer between high-order curvilinear meshes, and initialization of
multi-material simulations. We compare the quadrature scheme to commonly-used
quadrature schemes in the literature and show that our scheme is much more
efficient in terms of number of quadrature points used. We provide an
open-source implementation of the scheme in MATLAB as part of QuaHOG, a
software package for Quadrature of High-Order Geometries.
