含参量积分的分析性质及其应用.pdf

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1、-1-含参量积分的分析性质及其应用班级：11 数学与应用数学一班成绩：日期：2012 年 11 月 5 日-2-含参量积分的分析性质及其应用1.含参量正常积分的分析性质及应用1.1 含参量正常积分的连续性定理1 若二 元函 数),(yxf在矩 形区 域,dcbaR上 连续,则 函数x=dcdyyxf),(在a,b 上连续.例 1 设)sgn(),(yxyxf（这个函数在 x=y 时不连续）,试证由含量积分10),()(dxyxfyF所确定的函数在),(上连续.解因为10 x,所以当 y0,则 sgn(x-y)=1,即 f(x,y)=1.-1,xy 则yyydxdxyF01.21)1()(1,y

2、1 时,f(x,y)=-1,则101)1()(dxyF,即 F(x)=1-2y,0y1 又因).1(1)(lim),0(1lim10FyFFyyF(y)在 y=0 与 y=1 处均连续,因而 F(y)在),(上连续.例 2 求下列极限：（1）dxax11220lim;(2)2020coslimxdxx.解（1）因为二元函数22x在矩形域R=-1,1-1.1上连续,则由文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N

3、4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H

4、7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N

5、4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H

6、7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N

7、4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H

8、7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N

9、4W9X9-3-连续性定理得dxax1122在-1,1上连续.则1122110112201limlimdxxdxaxdxax.（2）因为二元函数axx cos2在矩形域2,2 2,0R上连续,由连续性定理得,函数202cosaxdxx在2,2上连续.则.38coslim2020220dxxaxdxx例 3 研究函数)(xFdxyxxyf1022)(的连续性,其中 f（x）在闭区间 0,1 上是正的连续函数.解对 任 意00y,取0,使00y,于 是 被 积 函 数22)(yxxyf在,1,000yyR上连续,根据含参量正常积分的连续性定理,则 F（y）在区间,00yy上连续,由0y的任意性知,

10、F（y）在),0(上连续.又因dxyxxyfdxyxxyfyF10221022)()()(,则 F（y）在)0,(上 连续.当 y=0 处0)(0yF.由于)(xf为0,1 上的正值连续函数,则存在最小值 m0.ymdxyxmydxyxxyfyF1arctan)()(10221022,从而04)(lim0myFy,但F(y)在 y=0 处不连续,所以 F（y）在),0(),(上连续,在 y=0 处不连续.定理 2 设二元函数 f(x,y)在区域 G=(x,y)|bxaxdyxc),()(上连续,其中 c(x),d(x)为a,b 上的连续函数,则函数 F(x,y)=)()(),(xdxcdyyx

11、f在a,b上连续.例 4 求12201limxdx.解记1221)(xdxI.由于2211,1,x都是和 x 的连续函数,由定理 2 知)(I在0处连续,所以41)0()(lim1020 xdxII.例 5 证明函数dxeyFyx0)(2)(在),(上连续.证明对),(y,令 x-y=t,可推得文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4

12、S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5

13、I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4

14、S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5

15、I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4

16、S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5

17、I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9-4-00000)(2)(22222yyttttyxdtedte

18、dtedtedxeyF.对于含多量正常积分02ytdte,由连续性定理可得02ytdte在),(上连续,则dxeyFyx0)(2)(在),(上连续.1.2 含参量正常积分的可微性定理 3 若函数fyx,与其偏导数xfyx,都在矩形区域 R=a,b*c,d上连续,则x=dyyxfdc),(在a,b 上可微,且dyyxfxdyyxfdxddcdc),(),(.定理 4 设fyx,xfyx,在 R=a,b*p,q上连续,cx,dx为定义在a,b 上其值含于 p,q 內的可微函数,则函数 Fx=)()(),(xdxcdyyxf在a,b 上可微,且).()(,()()(,(),()()()(xcxcxf

19、xdxdxfdyyxfxFxdxcx定理5 若函数fyx,及xfyx,都在 a,b;c,d上连续,同时在 c,d上)(ya及)(yb皆存在,并且 aa(y)b,a b(y)b(c yd),则)()()()()(),()(),(),(),()(ybyayybyayayyafybyybfdxyxfdxyxfdydyF.证明考虑函数 F(y)在c,d上任何一点处得导数,由于)()()(),(),(),()(3)()(21)()()()(000yFyFyFdxyxfdxyxfdxyxfyFyayaybybybyao.现在分别考虑)3,2,1)(iyFi在点0y处得导数.由定理 5 可得)()(0010

20、0),()(ybyaydxyxfyF.由于0)(02yF,所以dxyyyxfyyyFyyyFyFyFybybyyyyoyyo)()(0020220;2000),(lim)(lim)()(lim)(.应用积分中值定理),()()(lim)(00020yfyyybybyFyy.这里在)(yb和)(0yb之间.再注意到fyx,的连续性及 b(y)的可微性,于是得到),()()(00002yybfybyF.文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO

21、7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X

22、9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO

23、7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X

24、9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO

25、7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X

26、9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO

27、7H9B3S3Z8 ZL4S5N4W9X9-5-同样可以证明),()()(00003yyafyayF于是定理得证.例 6 设,sin)(2dxxyxyFyy求)(yF.解应用定理 5 有yyyyyyxdxyFyy223sin1sin2cos)(2yyyyyyxyy23sinsin2sin2yyy23sin2sin3.例 7 设)(xf在0 x的某个邻域 U上连续,验证当Ux时,函数dttftxnxnx)()()!1(1)(10（1）的 n 阶导数存在,且).()()(xfxn解由于（1）中被积函数)()(),(1tftxtxFn及其偏导数),(txFx在 U上连续,于是由定理 4 可得xnnx

28、fxxndttftxnnx012)()()!1(1)()(1()!1(1)(xndttftxn02.)()()!2(1同理xnnxfxxndttxnnx013)()()!1(1)(2()!2(1)(xndttftxn03.)()()!3(1如此继续下去,求得 k 阶导数为xknkdttftxknx01)(.)()()!1(1)(特别当1nk时有文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF

29、4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z

30、8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF

31、4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z

32、8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF

33、4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z

34、8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9-6-xndt

35、tfx0)1(,)()(于是).()()(xfxn例 8 计算积分.1)1ln(102dxxxI.解考虑含参量积分.1)1ln()(102dxxx显然,)1(,0)0(I且函数21)1ln(xx在 R=0,10,1 上满足定理 3 的条件,于是102.)1)(1()(dxxxx.因为),11(11)1)(1(222xxxxxx所以)()111(11101010222dxxdxxxdxx)1ln()1ln(21arctan1110102102xxx).1ln(2ln214112因此10)(d102)1ln(2ln21411d)1(arctan2ln21)1ln(810102)1(2ln82ln8

36、)1(2ln4.另一方面10),1()0()1()(d所以.2ln8)1(I1.3 含参量正常积分的可积性定理 6 若 fyx,在矩形区域R=ba,dc,上连续,则x和x分别在文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5

37、I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4

38、S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5

39、I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4

40、S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5

41、I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4

42、S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9-7-ba,和dc,上可积.其中x=dcyxf,dy,xba,x=bayxf,dy.这就是说：在fyx,连续性假设下,同时存在求积顺序不同的积分：dxdyyxfbadc,与dydxyx

43、fdcba,简 便 记 为dyyxfdxbadc,与dxyxfdydcba,前者表示 fyx,先对 y 求积然后对 x 求积,后者则表示先对 x 求积再对 y 求积.它们统称为累次积分或更确切地称为二次积分.由可积性的定理进一步指出,在 fyx,连续性假设下,累次积分与求积顺序无关,即若 fyx,在矩形区域 R=ba,dc,上连续,则dyyxfdxbadc,=dxyxfdydcba,.定理 7 若 fyx,在矩形区域R=ba,dc,上连续,gx在ba,上可积,则作为 y 的函数dxxgyxfba,在dc,上连续,且dyyxfdxxgdccba,=dxxgyxfdydcba,.注意 推论中闭区间

44、dc,可以换成开区间或无穷区间,因为可积性定理是由连续性推得的,连续性是局部性质.例 9 求 I=dxxxxab10ln（ba0）.解由xxxdyxabbayln得 I=dxdyxbay10=10baydyxdx,因为yxyxf,在矩形区域ba,1,0上连续,由定理可得 I=dxxdybay10=dyyba11=lnab11.例 10 试求累次积分dyyxyxdx101022222与101022222dxyxyxdy,并指出它们为什么与定理的结果不符.解：dyyxyxdx101022222=dxdyyxyx101022222=dxyxyxdyyxdy1010102222222=dxyxydyx

45、dy10101022221=dxx10211=0arctan1arctan=4.101022222dxyxyxdy=dxxyxydy101022222,由dyyxyxdx101022222=4,同 理 可 得文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4

46、W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7

47、 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4

48、W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7

49、 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4

50、W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7 HO7H9B3S3Z8 ZL4S5N4W9X9文档编码:CF4D8Q5I4H7