定积分的应用论文

学号：

本科毕业论文

学 院 专 业 年 级 姓 名 论文题目 定积分的若干应用 指导教师 薛艳昉 职称 讲师

2013年5月16日

目 录

摘 要 ····························································································· 1 关键词 ····························································································· 1 Abstract ·························································································· 1 Keywords ························································································ 1 0 前 言 ························································································· 1 1 定积分在数学中的应用 ································································· 1

1.1 曲边梯形面积的求法 ·································································· 1 1.2 扇形面积的求法 ········································································ 3 1.3 立体图形的体积的求法 ······························································· 3 1.4 由截面面积求旋转体的体积 ························································· 4 1.5 求弧长的方法 ··········································································· 5 1.6 由微分法求旋转曲面的面积 ························································· 6 1.7 利用定积分对数列求和 ······························································· 7 1.8 利用定积分进行因式分解、化简代数式 ·········································· 7 1.9 利用定积分证明不等式 ······························································· 8

2 定积分在物理中的应用 ································································· 9

2.1 液体静压力 ·············································································· 9 2.2 引力问题 ················································································· 9 2.3 功与平均功率 ·········································································· 10

3 定积分在经济中的应用 ································································ 12

3.1 最大利润问题 ·········································································· 12 3.2 资金的现值、终值与投资问题 ····················································· 12

参考文献 ························································································ 13

定积分的若干应用

姓名： 学号：

数学与信息科学学院 数学与应用数学 指导老师： 职称：讲师

摘 要：本文通过定积分中微元法的思想，讨论了定积分在数学、物理学以及经济学中的若干应用，包括立体图形的体积的求法、不等式的证明、液体静压力、引力问题、最大利润问题等．

关键词：定积分；微分法；弧长

Some Application of Integral

Abstract：In this paper，we discuss some application of integral in mathematics， physics and economics through the thought infinitesimal method，including the volume of three-dimensional，graphics for France Inequality，hydrostatic pressure，gravity issues，the maximum profit problms．

Keywords：definite integral；differential method；arc length

0 前言

微积分是数学的一个重要分支，它是科学技术以及自然科学的各个分支中被广泛应用的最重要的数学工具之一，如复杂图形的研究，求数列极限等问题，在物理学方面液体静压力，引力等的研究，以及在经济学中利润投资等问题的决策都需要定积分的知识．以下将介绍定积分在这三方面的若干应用实例．

1 定积分在数学中的应用

1.1 曲边梯形的面积的求法?1?

设f为闭区间[a,b]上连续函数，且f(x)?0，由曲线y?f(x)，直线x?a，

x?b以及x轴所围成的平面图形．

下面讨论该曲边梯形的面积．我们在初等数学中，圆的面积是用一系列边数无限增加的内接（或外切）正多边形的面积的极限来定义的，现在我们仍用类似的方法来定义曲边梯形的面积．根据这一思想我们可以得到曲边梯形的面积公式为s??f(x)dx．

ab 1

由此可知，由上下两条连续曲线y1?f(x)，y2?g(x)以及直线x?a和直线

x?b(a?b)所围的平面图形的面积，它的计算公式为A???g(x)?f(x)?dx．

ab 例1 求抛物线y2?x与直线x?2y?3所围成的平面图形的面积．

解 设抛物线与直线的交点P(?1,1)与Q(9,3)．用直线x?1把图形分为左、右两个部分，应用公式分别求得它们的面积为

A1=?[x?(?x)dx?2?0110xdx?4， 3A2=?(x?19x?328)dx?． 23所以A?A1?A2?42832??． 333 设曲线C由参数方程

x=x(t)，y?y(t)，t?[?,?]

给出，在[?,?]上y(t)连续，x(t)连续可微且x'?t??0（对于y(t)连续可微y'?t??0的情形可类似地讨论）．记a=x(?)，b=x(?)，(a?b或b?a)，则由曲线C及直线

x?a,x?b和x轴所围的图形，其面积计算公式为

A??|y(t)x?(t)|dt．

?? 如果由参数方程表示的曲线是封闭的，那么由曲线自身所围的图形的面积为

A???y(t)x?(t)dt．

?x2y2 例2 求椭圆2?2?1所围的面积．

ab 解 化椭圆方程为参数方程

x=acost，y=bsint，t?[0,2?]．

则可求得椭圆围面积

A=|?0bsint(acost)`dt|

=ab2??2?0sin2tdt=?ab．

显然，当a?b?r时，这就等于圆面积?r2

2

1.2 扇形面积的求法?2?

设曲线C由极坐标方程

r=r(?)，??[?,?]

给出，其中r(?)在[?,?]上连续，????2?．由曲线C与两条射线???,???所围成的平面图形，通常也称为是扇形．此扇形的面积的计算公式为

1?2A=?r(?)d?．

2? 例3 求双纽线r2=a2cos2?所围成的平面图形的面积． 解 因为r2?0，所以?的取值范围为[-?3?5???,]与[,]．由图形的对称性得

444,41A=4??04a2cos2?d?=a2．

21.3 立体图形的体积的求法?3?

设S是三维空间中一立体，它夹在垂直于x轴的两平面x?a与x?b之间

?a?b?．为方便起见称S为位于[a,b]上的立体．若在任意一点x?[a,b]处作垂直于

x轴的平面，它截得S的截面面积显然是x的函数，记为A(x)，x??a,b?，并称之为S的截面面积函数．

设截面面积函数A(x)是?a,b?上的一个连续函数．对[a,b]作分割

T：a?x0?x1?...?xn?b．

过各个分点作垂直于x轴的平面x?xi，i?1,2,...,n，它们把S切割成n个薄片．设

A(x)在每个小区间?i??xi?1,xi?上的最大、小值分别为Mi与mi，那么每一薄片的

体积?Vi满足

mi?xi??Vi?Mi?xi．

于是，S的体积V???Vi满足

i?1n 3

本文来源：https://www.bwwdw.com/article/4imf.html