Simpson's Rule is an accurate numerically stable method of approximating a definite integral using a quadrature with three points, obtained by integrating the unique quadratic that passes through these points. The error term in the method is a function of the fourth derivative of the integrand. Therefore, it is easy to see that the method is exact for cubics, since the fourth derivative of a cubic is zero, and there is no error. The error analysis uses Taylor series. In our simple proof, we will use ordinary integration techniques.
"Simpson's Rule is Exact for Cubics: A Simple Proof." Missouri J. Math. Sci. 17 (2) 100 - 105, Spring 2005. https://doi.org/10.35834/2005/1702100