Polynomial integration on regions defined by a triangle and a conic

Abstract: We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type [\iint_{T \cap {f\ge0}} \phi_1\phi_2 \, dx\,dy] for quadratic polynomials $f,\phi_1,\phi_2$ on a plane triangle $T$. The naive approach would involve consideration of the many possible shapes of $T\cap{f\geq0}$ (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables separately. Our solution involves partitioning the triangle into smaller triangles on which integration is much simpler.