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CollegeofmathematicsMarch27,20207.3二重积分的应用ApplicationsofDoubleIntegralsCollegeofmathematicsMarch27,20207.2.1二重积分的元素法CollegeofmathematicsMarch27,2020上一页|首页|下一页将定积分的元素法推广到二重积分可得二重积分的元素法CollegeofmathematicsMarch27,2020上一页|首页|下一页若要计算的某个量U对于闭区域D具有可加性DUdUdddyxf),(),(yx并且在闭区域D内任取一个直径很小的闭区域时,相应地部分量可近似地表示为的形式,其中在内.则所求量的积分表达式为dyxf),(dU称为所求量U的元素,记为,(,)Dfxyd二重积分的元素法(即当闭区域D分成许多小闭区域时,所求量U相应地分成许多部分量,且U等于部分量之和),CollegeofmathematicsMarch27,20207.3.2曲面的面积TheAreaofaSurfaceCollegeofmathematicsMarch27,2020上一页|首页|下一页平面的面积:0AxByCzD求平面在有界闭区域D上的那一块的面积ADA已知求ACollegeofmathematicsMarch27,2020上一页|首页|下一页则cosAcosA设平面与z=0的夹角为DAAACollegeofmathematicsMarch27,2020上一页|首页|下一页:0AxByCzD平面与z=0的夹角余弦为222cosCABCn{,,}ABC与k夹角余弦所以cosA222ABCCDACollegeofmathematicsMarch27,2020上一页|首页|下一页曲面方程:(,)zfxy(,)xyD:D有界闭区域求曲面的面积ADCollegeofmathematicsMarch27,2020上一页|首页|下一页dA(,)xydA(,)zfxy切平面的法矢:n{,,1}xyff切平面与z=0的夹角余弦:221cos1xyff(n与k的夹角余弦)CollegeofmathematicsMarch27,2020上一页|首页|下一页221cos1xyffdA(,)xydAdAcosd221xydAffd面积元素ADdA221xyDffdCollegeofmathematicsMarch27,2020上一页|首页|下一页221xyDAffdD曲面面积其中D是曲面在坐标面z=0上的投影区域求曲面面积的步骤:(1)求曲面在坐标面z=0上的投影区域D(2)在区域D上计算二重积分:221xyDAffdCollegeofmathematicsMarch27,2020上一页|首页|下一页设曲面的方程为:(,)yhzx曲面面积公式为:221zxyyzxDAdzdx设曲面的方程为:),(zygx曲面面积公式为:221yzxxyzDAdydz同理可得),(zygxyzDzxD),(xzhyCollegeofmathematicsMarch27,2020上一页|首页|下一页Example22zxy22:Dxyxwith(plots):zuimian:=implicitplot3d(x^2+y^2=z^2,x=-2..2,y=-2..2,z=0..1,color=yellow,grid=[20,20,20]):zhumian:=implicitplot3d(x^2+y^2=x,x=-2..2,y=-2..2,z=0..1,color=green,grid=[20,20,20]):x_axis:=plot3d([u,0,0],u=-2..2,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=-2..2,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..1.2,v=0..0.01,thickness=2):display(zuimian,zhumian,x_axis,y_axis,z_axis,orientation=[23,66],scaling=constrained);投影区域Projection22xyx22xyx1CollegeofmathematicsMarch27,2020上一页|首页|下一页22zxy22:Dxyx22zxxxy22zyyxy221()()zzxy2222221xyxyxy2A221()()Dzzdxy2Dd22422xyx1CollegeofmathematicsMarch27,2020上一页|首页|下一页Example222zaxyzh(0)haazhD222zaxyzh球冠在xOy面上的投影区域:2222:DxyahCollegeofmathematicsMarch27,2020上一页|首页|下一页222zaxy2222:Dxyah222zxxaxy221()()zzxy222222221xyaxyaxy222zyyaxy222aaxyazhDCollegeofmathematicsMarch27,2020上一页|首页|下一页A221()()Dzzdxy2222:Dxyah221()()zzxy222aaxy222Dadaxy2222200ahadrdrar222202(1)[]ahar2()aah2aH()HahazhDCollegeofmathematicsMarch27,2020上一页|首页|下一页2AaH2()AaahazhH半球面面积:A0lim2()haah22a球面面积:24AaaCollegeofmathematicsMarch27,20207.3.3平面薄片的重心TheCenterofMassofaLaminaCollegeofmathematicsMarch27,2020上一页|首页|下一页质点:P(x,y)质量:m质点P对x轴的静力矩:xMymy(,)Pxyx质点P对y轴的静力矩:yMxmCollegeofmathematicsMarch27,2020上一页|首页|下一页平面薄片占据区域D(,)xy面密度:(,)xy取一小块薄片:dd位于(x,y)近似地看成质点小薄片质量:(,)dMxyd质量元素D薄片质量:DMdM(,)DxydCollegeofmathematicsMarch27,2020上一页|首页|下一页yxd小薄片对x轴的静力矩:(,)xdMyxyd对y轴的静力矩元素小薄片对y轴的静力矩:(,)ydMxxyd对x轴的静力矩元素CollegeofmathematicsMarch27,2020上一页|首页|下一页yxd薄片对x轴的静力矩:xxDMdM(,)Dyxyd薄片对y轴的静力矩:yyDMdM(,)DxxydThemomentaboutthex-axisThemomentaboutthey-axisCollegeofmathematicsMarch27,2020上一页|首页|下一页(,)xDMxyyd(,)yDMxyxd薄板的重心坐标:(,)xyyMxM(,)(,)DDxxydxydxMyM(,)(,)DDyxydxydThecenterofthemass(,)xyCollegeofmathematicsMarch27,2020上一页|首页|下一页DDxdxdDDydyd若薄片有均匀密度:(常数)薄板的重心坐标:DDxdxdDDxdd1DxdCollegeofmathematicsMarch27,2020上一页|首页|下一页DDydydDDydd1Dyd此时的中心成为形心1Dxxd1Dxxd1DyydCollegeofmathematicsMarch27,2020上一页|首页|下一页例3.424yx2yx是常数求形心作图implicitplot(y^2=4*x,x=-0.2..1.2,y=-0.5..2.2,thickness=3,scaling=constrained);交点:(0,0)(1,2)24yx2yxDdD1202xxdxdy2yx102()xxdx13CollegeofmathematicsMarch27,2020上一页|首页|下一页2yxD2yxDxd1202xxdxxdy102()xxxdx215Dyd1202xxdxydy1201()2xxdx13CollegeofmathematicsMarch27,2020上一页|首页|下一页13215132yx2yx211532511331形心:(,)xy2(,1)5DxdDyd1Dxxd1DyydCollegeofmathematicsMarch27,2020上一页|首页|下一页例3.3自学由对称性,重心在x轴上:0yCollegeofmathematicsMarch27,20207.3.4平面薄片的转动惯量TheMomentofInertiaofaLaminaCollegeofmathematicsMarch27,2020上一页|首页|下一页质点:P(x,y)质量:m质点P对x轴的转动惯量:2xIymy(,)Pxyx质点P对y轴的静力矩:2yIxmy:转动半径x:转动半径CollegeofmathematicsMarch27,2020上一页|首页|下一页质点P对原点O的转动惯量:22()OIxym转动半径22xyO(,)Pxy22xyCollegeofmathematicsMarch27,2020上一页|首页|下一页平面薄片占据区域D(,)xy面密度:(,)xy取一小块薄片:dd位于(x,y)近似地看成质点小薄片质量:(,)dMxyd质量元素DCollegeofmathematicsMarch27,2020上一页|首页|下一页yxd小薄片对x轴的转动惯量:2(,)xdIyxyd对y轴的转动惯量元素小薄片对y轴的转动惯量:2(,)ydIxxyd对x轴的转动惯量元素对O的转动惯量元素小薄片对原点O的转动惯量:22()(,)OdIxyxydCollegeofmathematicsMarch27,2020上一页|首页|下一页yxd薄板对x轴的转动惯量:xIxDdI2(,)Dxyyd薄板对y轴的转动惯量:yIyDdI2(,)DxyxdThemomentofinertiaaboutthex-axisThemomentofinertiaaboutthey-axis2(,)xdIyxyd2(,)ydIxxydCollegeofmathematicsMarch27,2020上一页|首页|下一页薄板对原点O的转动惯量:OIODdI22(,)()DxxydyThemomentofinertiaaboutthe
本文标题:7.3-二重积分的应用
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