A Version of Simpson's Rule for Multiple Integrals

Let M(f) denote the Midpoint Rule and T(f) the Trapezoidal Rule for estimating integral_a^b f(x) dx. Then Simpson's Rule = tM(f) + (1-t)T(f), where t = 2/3. We generalize Simpson's Rule to multiple integrals as follows. Let D be some polygonal region in R^n, let P_0,...,P_m denote the vertices of D, and let P_(m+1) = center of mass of D. Define the linear functionals M(f) = Vol(D)f(P_(m+1)), which generalizes the Midpoint Rule, and T(f) = Vol(D)(1/(m+1))sum(f(P_j), j = 0,...,m), which generalizes the Trapezoidal Rule. Finally, our generalization of Simpson's Rule is given by the cubature rule(CR) L_t = tM(f) + (1-t)T(f), for t in [0,1]. We choose t, depending on D, so that L_t is exact for polynomials of as large a degree as possible. In particular we derive CRs for the n simplex and unit n cube.