定积分的应用论文

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日学号：目录摘要·····························································································1关键词·····························································································1Abstract··························································································1Keywords························································································10前言·························································································11定积分在数学中的应用·································································11.1曲边梯形面积的求法··································································11.2扇形面积的求法········································································31.3立体图形的体积的求法·······························································31.4由截面面积求旋转体的体积·························································41.5求弧长的方法···········································································51.6由微分法求旋转曲面的面积·························································61.7利用定积分对数列求和·······························································71.8利用定积分进行因式分解、化简代数式··········································71.9利用定积分证明不等式·······························································82定积分在物理中的应用·································································92.1液体静压力··············································································92.2引力问题·················································································92.3功与平均功率··········································································103定积分在经济中的应用································································123.1最大利润问题··········································································123.2资金的现值、终值与投资问题·····················································12参考文献························································································131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SomeApplicationofIntegralAbstract：Inthispaper，wediscusssomeapplicationofintegralinmathematics，physicsandeconomicsthroughthethoughtinfinitesimalmethod，includingthevolumeofthree-dimensional，graphicsforFranceInequality，hydrostaticpressure，gravityissues，themaximumprofitproblms．Keywords：definiteintegral；differentialmethod；arclength0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],[ba上连续函数，且0)(xf，由曲线)(xfy，直线ax，bx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badxxfs)(．2由此可知，由上下两条连续曲线)(1xfy，)(2xgy以及直线ax和直线bx)(ba所围的平面图形的面积，它的计算公式为badxxfxgA)()(．例1求抛物线xy2与直线32yx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)3,9(．用直线1x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1A=342)([1010dxxdxxx，2A=328)23(91dxxx．所以3323283421AAA．设曲线C由参数方程x=)(tx，)(tyy，t],[给出，在[],上)(ty连续，)(tx连续可微且0'tx（对于)(ty连续可微0'ty的情形可类似地讨论）．记a=)(x，b=)(x，)(abba或，则由曲线C及直线bxax,和x轴所围的图形，其面积计算公式为dttxtyA|)()(|．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dttxtyA)()(．例2求椭圆12222byax所围的面积．解化椭圆方程为参数方程x=tacos，y=tbsin，t]2,0[．则可求得椭圆围面积A=|20)`cos(sindttatb|=ab202sintdt=ab．显然，当rba时，这就等于圆面积2r31.2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A=dr)(212．例3求双纽线r2=a2cos2所围成的平面图形的面积．解因为r02，所以的取值范围为[-4,,4]与[45,43]．由图形的对称性得A=4da2cos21402=2a．1.3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ax与bx之间ba．为方便起见称S为位于],[ba上的立体．若在任意一点x],[ba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)(xA，bax,，并称之为S的截面面积函数．设截面面积函数)(xA是ba,上的一个连续函数．对],[ba作分割T：bxxxan...10．过各个分点作垂直于x轴的平面ixx，ni,...,2,1，它们把S切割成n个薄片．设)(xA在每个小区间iiixx,1上的最大、小值分别为iM与im，那么每一薄片的体积iV满足iiiiixMVxm．于是，S的体积niiVV1满足4iniiiniixMVxm11．因为)(xA为连续函数，从而在],[ba上可积，所以当T足够小时，能使iniiiiniixmMx11，其中为任意小的正数．由此知道iniiTxMV10lim（或iniiTxm10lim）＝iniiTxAi10lim，其中iiMA（或im），所以有dxxAVba．例4求椭球面1222222czbyax所围的立体的面积．解以平面0xx|)(|0ax截椭球面，得椭圆1)1()1(2202222022axczaxby，所以截面面积函数为)(xA=],[),1(22aaxaxbc，于是求得椭球的体积为：V=dxaxbcaa)1(22=abc34．1.4由截面面积求旋转体的体积设f是],[ba的连续函数，S是有平面图形0bxaxfy|,)(|||绕x轴一周所得的旋转体，那么易知截面面积函数为)(xA=],[,)]([2baxxf．由V=badxxA)(可知，旋转体S的体积公式为5dxxfVba2)]([．例5求由圆x)0()(222RrrRy绕x轴旋转一周所得环状立体的体积．解)0()(222RrrRyx的上、下半圆分别为：222xrRxfy，rx．221xrRxfy，rx．所以圆环体的截面面积函数是2122)]([)]([)(xfxfxA224xrr，rrx,．由此可得圆环体的体积为dxxrRVr0228=Rr222．1.5求弧长的方法由弧长的概念可知弧长S=dttytx)()(22．若曲线C由直线坐标方程],[),(baxxfy表示，若把它看作参数方程，即为],[),(,baxxfyxx．所以当)(xf在],[ba上连续可微时，此曲线即为一光滑曲线．这时弧长公式为dxxfSba)(12．又若曲线C由极坐标方程],[),(rr表示，把它化为参数方程，则为)cos()(rx，)sin()(ry，,．由于sin)(cosrrx，cos)(sinrry，2222rryx，6因此当r在,上连续，且r和r不同时为零时，此极坐标曲线为一光滑曲线．这时弧长公式为drrS)()(22．例6求摆线x=a（t-tsin），