  This work presents spectral, mesh-free, Green's theorem-based numerical
quadrature schemes for integrating functions over planar regions bounded by
rational parametric curves. Our algorithm proceeds in two steps: (1) We first
find intermediate quadrature rules for line integrals along the region's
boundary curves corresponding to Green's theorem. (2) We then use a high-order
quadrature rule to compute the numerical antiderivative of the integrand along
a coordinate axis, which is used to evaluate the Green's theorem line integral.
We present two methods to compute the intermediate quadrature rule. The first
is spectrally accurate (it converges faster than any algebraic order with
respect to number of quadrature points) and is relatively easy to implement,
but has no guarantee of polynomial exactness. The second guarantees exactness
for polynomial integrands up to a pre-specified degree k with an a priori-known
number of quadrature points and retains the convergence properties of the
first, but is slightly more complicated. The quadrature schemes have
applications to computation of geometric moments, immersogeometric analysis,
conservative field transfer between high-order meshes, and initialization of
simulations with rational geometry. We compare the quadrature schemes produced
using our method to other methods in the literature and show that they are much
more efficient both in terms of number of quadrature points and computational
time.
