求某函数的定积分时，在多数情况下，被积函数的原函数很难用初等函数表达出来，因此能够借助微积分学的牛顿-莱布尼兹公式计算定积分的机会是不多的。另外，许多实际问题中的被积函数往往是列表函数或其他形式的非连续函数，对这类函数的定积分，也不能用不定积分方法求解。由于以上原因，数值积分的理论与方法一直是计算数学研究的基本课题。对微积分学作出杰出贡献的数学大师，如I.牛顿、L.欧拉、C.F.高斯、拉格朗日等人都在数值积分这个领域作出了各自的贡献，并奠定了这个分支的理论基础。(For a definite integral function, in most cases, the primary function of the integrand is difficult to be expressed by elementary function, so it can use the Newton Leibniz formula of calculus integral calculation is a rare chance. In addition, the integrand in many practical problems is often a list function or other form of discontinuous function, and the definite integral of such functions can not be solved by indefinite integral method. For these reasons, the theory and method of numerical integration has been the basic subject of computational mathematics. Master of mathematics made outstanding contributions to the calculus, such as I. Newton, L. C.F. Gauss, Euler, Lagrange and others in the field of numerical integration to make their respective contributions, and laid the theoretical foundation of this branch.)