An Elementary Proof of Error Estimates for the Trapezoidal Rule
An Elementary Proof of Error Estimates for the Trapezoidal Rule
复制标题
梯形规则误差估计的基本证明
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
C. Neugebauer
中科院分区：
文献类型：
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作者：
D. Cruz;C. Neugebauer
(See, for instance, Stewart [7].) There are two problems, however, with this result. First, calculus books generally omit the proof, and instead refer the reader to an advanced text on numerical analysis. In such books the trapezoidal rule is usually derived as a corollary to a more general result for Newton-Cotes quadrature methods, and the proof, depending on polynomial approximation, is generally not accessible to calculus students. (See, for example, Ralston [5].) Second, the error estimate given by (1) is not applicable to such well-behaved functions as x3/2 and x /2 on [0, 1], since neither has a bounded second derivative. In other words, you can use the trapezoidal rule to approximate their integrals, but for a given n you have no idea, a priori, how good the approximation is. In this note we give an elementary proof of inequality (1). The key idea is to use integration by parts "backwards." The argument is straightforward and should be readily understood by students in second-semester calculus. This approach is not new and goes back to Von Mises [8] and Peano [4]. (See also Ghizzetti and Ossicini [3].) However, our exact proof is either new or long forgotten-see Cruz-Uribe and Neugebauer [2] for a survey of the literature. Our proof has two other advantages. First, we can omit the assumption that f" is continuous, and replace it with the weaker assumption that it is (Riemann) integrable. Second, we can adapt our proof to give estimates for functions that do not have bounded second derivatives, such as x3/2 and x12.
