在前两部分中，根据一元函数的调查研究方法，我们将经典的中点原则 和梯形原则扩展到二元函数上，并给出了二元函数上估计积分值的新的带有 系数θ的中点-梯形原则。 在第三节中我们利用带有积分型余项的泰勒公式来进行误差分析，首先 对一元函数进行估计，再将二元函数用同样的方法，利用多元函数的泰勒展 开进行估计，最终得到了二元函数的重点-梯形原则的误差估计，并且给出了 该原则的应用。

In the first two parts, following the methods of researching functions of one variable, we expand the modern midpoint rule and trapezoid rule into the functions of two variables, and give a new midpoint-trapezoid rule with a coefficient θ on functions of two variables to estimate the integration. In the third part, we discover the error analysis through the Taylor formula with integral remainder term. Estimate the error of the functions of one variable. Then approximate the integration of functions of two variables through the Taylor expansion of multivariate functions, get the error estimation at last. The application is also given.