Simpson's Rule Revisited

Generalized Alomari functionals

We consider a generalized form of certain integral inequalities given by Guessab, Schmeisser and Alomari. The trapezoidal, mid point, Simpson, Newton-Simpson rules are obtained as special cases. Also, inequalities for the generalized Alomari functional in terms of the $n$-th order modulus, $n=\overline{1,4}$, are given and applied to some known quadrature rules.

辛普森法则的多重积分版本

A Version of Simpson's Rule for Multiple Integrals

本文将辛普森法则推广到多重积分，通过中点法则与梯形法则的凸组合，并选择参数t使求积法则对高次多项式精确，推导了n单纯形和单位n立方体的求积公式。

基本求积公式的简单推导

Simple derivation of basic quadrature formulas

本文通过分部积分法，简洁地证明了中点法则、梯形法则和辛普森法则的误差估计，并推广到包含端点一阶导数的修正梯形法则。