积分中值定理的推广及应用

海南大学

毕业论文（设计）

题目：积分中值定理的推广及应用

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学院：信息科学技术学院

系别：数学系

专业：信息与计算科学

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完成日期：年月日

摘要

本论文讲述的主要内容是积分中值定理及其应用，我们将它主要分为以下几个方面：积分中值定理、积分中值定理的推广、积分中值定理中值点ξ的渐进性，积分中值定理的应用。

我们讨论了定积分中值定理、第一积分中值定理、第二积分中值定理，而且还给出了这些定理的详细证明过程。在此基础上，我们还讨论了在几何形体Ω上的黎曼积分第一中值定理，它使得积分中值定理更加一般化，此情形对于讨论一般实际问题有很显著作用。

在积分中值定理的推广方面，我们由最初的在闭区间[,]

f x的积分中值

a b讨论函数()

定理情形转换为在开区间(,)

f x上的积分中值定理，这个变化对于解决一

a b上讨论函数()

些实际的数学问题更为方便。不仅如此，我们还将几何形体Ω上的黎曼积分第一中值定理推广到第一、第二曲线型积分中定理和第一、第二曲面型积分中值定理情形。

有关ξ点的渐进性，我们对第一积分中值定理的ξ点的做了详细的讨论，给出详细清楚的证明过程。而第二积分中值定理的渐进性问题只证明了其中的一种情形，其它证明过程只做简要说明。

对于应用，我们给出了一些较简单的情形如估计积分值，求含有定积分的极限，确定积分号，比较积分大小，证明函数的单调性还有对阿贝尔判别法和狄理克莱判别法这两个定理的证明。

关键词：积分中值定理；推广；应用；渐进性

Abstract

The main content of this paper are the mean-value theorem and its application, it will be mainly divided into the following respects: integral mean-value theorem, the generalation of integral mean-value theorem, the asymptotic property of the “intermediate point”of integral median point, the application of integral mean-value theorem.

We have discussed the definite integral mean-value theorem, the first mean value theorem, the second integral mean-value theorem, and have given a detailed proof of these theorems process. On this basis, we also have discussed the Riemann first integral mean-value theorem on the geometryΩ. It makes the integral mean-value theorem is more general, the case has a significant role in the discussion of practical issues in general.

In the promotion of integral mean value theorem, we have discussed the integral mean-value theorem of function ()

a b in the case of

f x in the initial closed interval [,]

discussing it in the open interval(,)

a b, the change has more convenience in solving some practical mathematical problem. In addition, we will promote the Riemann first integral mean-value theorem on the geometryΩto the situation of the first and second type curve in integral theorem and The second type surface integral mean-value theorem.

About the Progressive of ξpoint, we have discussed the ξpoint of the mean-value theorem in detail and give clear proof of the process. While the gradual issues of the second integral mean value theorem has been demonstrated one of these situations. And the other process of proving has been expressed in brief.

According to application,we presented a simple situation, for example, estimate integral value ,solve the limits of definite integral, define integral sign, compare the magnitude of integral value, prove the monotonic of function and Abel test and Dirichlet test

Key words: integral mean-value; theorem promotion ;apply;progressive