# Simple Formula for Integration of Polynomials on a Simplex

## TL;DR

The paper presents a straightforward formula that simplifies the integration of polynomials over a simplex by reducing it to evaluations at specific points, enhancing computational efficiency in finite element methods.

## Contribution

It introduces a novel, simple formula for integrating polynomials on a simplex, reducing the problem to evaluating homogeneous polynomials at specific points.

## Key findings

- Simplifies polynomial integration on simplexes.
- Reduces integration to polynomial evaluations at specific points.
- Applicable in finite element and related methods.

## Abstract

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.