  We show that integrating a polynomial of degree t on an arbitrary simplex
(with respect to Lebesgue measure) reduces to evaluating t homogeneous
polynomials of degree j = 1, 2,. .. , t, each at a unique point $\xi$ j of the
simplex. This new and very simple formula can be exploited in finite (and
extended finite) element methods, as well as in other applications where such
integrals are needed.
