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.

中文翻译：

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多项式在单纯形上积分的简单公式

我们表明，在任意单纯形（相对于勒贝格测度）上对 t 次多项式进行积分可以简化为评估 t 次 j = 1, 2, 的齐次多项式。.. , t, 每个都在单纯形的唯一点 $\xi$ j 处。这个新的非常简单的公式可以在有限（和扩展的有限）元方法中使用，也可以在需要这种积分的其他应用中使用。