7.4.2三重積分的計(jì)算

II.利用柱面坐標(biāo)和球面坐標(biāo)計(jì)算三重積分TripleIntegralsinCylindricalandSphericalCoordinates1柱面坐標(biāo)CylindricalCoordinates柱面坐標(biāo)＝極坐標(biāo)＋豎坐標(biāo)柱面坐標(biāo)2規(guī)定：3柱面坐標(biāo)的三組坐標(biāo)面常數(shù)圓柱面：常數(shù)柱面坐標(biāo)因此得名4常數(shù)半平面5常數(shù)平面with(plots):x_axis:=plot3d([u,0,0],u=0..2,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..2,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..11,v=0..0.01,thickness=2):forifrom1to40dopingmian[i]:=plot3d([t,s,i/4],t=0..2,s=0..2,color=blue,style=patchnogrid):od:pingmian:=display(seq(pingmian[i],i=1..40),insequence=true):display(pingmian,x_axis,y_axis,z_axis,orientation=[24,67]);6柱面坐標(biāo)的坐標(biāo)面7柱面坐標(biāo)的坐標(biāo)面動(dòng)畫runinMaplewith(plots):x_axis:=plot3d([u,0,0],u=0..2,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..2,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..2.3,v=0..0.01,thickness=2):forifrom1to40dozhumian[i]:=plot3d([2*i/40,theta,z],theta=0..2*Pi,z=0..2,color=green,style=patch,coords=cylindrical):pingmian[i]:=plot3d([x,y,2*i/40],x=-2..2,y=-2..2,color=blue,style=patchnogrid):banpingmian[i]:=plot3d([r,2*Pi*i/40,z],r=0..3,z=0..2,color=brown,style=patchnogrid,coords=cylindrical):od:banpingmian:=display(seq(banpingmian[i],i=1..40),insequence=true):pingmian:=display(seq(pingmian[i],i=1..40),insequence=true):zhumian:=display(seq(zhumian[i],i=1..40),insequence=true):display(pingmian,banpingmian,zhumian,x_axis,y_axis,z_axis,orientation=[24,67]);8用以上三組坐標(biāo)面劃分區(qū)域兩張相鄰的水平面兩個(gè)相鄰的圓柱面兩張相鄰的半平面9with(plots):x_axis:=plot3d([u,0,0],u=0..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3.3,v=0..0.01,thickness=2):banpingmian1:=plot3d([r,Pi/6,z],r=0..3,z=0..2,coords=cylindrical,style=patch,color=green):banpingmian2:=plot3d([r,Pi/3,z],r=0..3,z=0..2,coords=cylindrical,style=patch,color=green):zhumian1:=plot3d([1,theta,z],theta=0..2*Pi,z=0..2,coords=cylindrical,style=patch,color=red):zhumian2:=plot3d([2,theta,z],theta=0..2*Pi,z=0..2,coords=cylindrical,style=patch,color=red):pingmian1:=plot3d([r,theta,1],r=0..3,theta=0..2*Pi,coords=cylindrical,style=patch,color=blue):pingmian2:=plot3d([r,theta,1.5],r=0..3,theta=0..2*Pi,coords=cylindrical,style=patch,color=grey):display(pingmian1,pingmian2,banpingmian1,banpingmian2,zhumian1,zhumian2,x_axis,y_axis,z_axis,orientation=[5,60]);10with(plots):x_axis:=plot3d([u,0,0],u=0..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3.3,v=0..0.01,thickness=2):banpingmian1:=plot3d([r,Pi/6,z],r=1..2,z=1..1.5,coords=cylindrical,style=patchnogrid,color=green):

banpingmian2:=plot3d([r,Pi/3,z],r=1..2,z=1..1.5,coords=cylindrical,style=patchnogrid,color=green):zhumian1:=plot3d([1,theta,z],theta=Pi/6..Pi/3,z=1..1.5,coords=cylindrical,style=patchnogrid,color=red):zhumian2:=plot3d([2,theta,z],theta=Pi/6..Pi/3,z=1..1.5,coords=cylindrical,style=patchnogrid,color=red):pingmian1:=plot3d([r,theta,1],r=1..2,theta=Pi/6..Pi/3,coords=cylindrical,style=patchnogrid,color=blue):pingmian2:=plot3d([r,theta,1.5],r=1..2,theta=Pi/6..Pi/3,coords=cylindrical,style=patchnogrid,color=blue):display(banpingmian1,banpingmian2,zhumian1,zhumian2,pingmian1,pingmian2,x_axis,y_axis,z_axis,orientation=[10,70]);11常見曲面的柱面坐標(biāo)方程直角坐標(biāo)方程柱面坐標(biāo)方程半球面12直角坐標(biāo)方程柱面坐標(biāo)方程圓錐面旋轉(zhuǎn)拋物面with(plots):qumian:=implicitplot3d(z=x^2+y^2,x=-2..2,y=-2..2,z=0..2,color=yellow,grid=[15,15,15]):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..3,v=0..0.01,thickness=2):display(qumian,x_axis,y_axis,z_axis,orientation=[23,66],scaling=constrained);with(plots):x_axis:=plot3d([u,0,0],u=0..1.2,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..1.2,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..1.5,v=0..0.01,thickness=2):zuobiaoxi:=display(x_axis,y_axis,z_axis):qumian:=plot3d([u*cos(theta),u*sin(theta),u],u=0..1,theta=0..2*Pi,numpoints=800,color=green):display(qumian,zuobiaoxi,orientation=[35,60]);13直角坐標(biāo)方程柱面坐標(biāo)方程圓柱面圓柱面14常見曲面的柱面坐標(biāo)方程直角坐標(biāo)方程柱面坐標(biāo)方程半球面圓錐面旋轉(zhuǎn)拋物面圓柱面圓柱面15例螺旋面

柱面坐標(biāo)方程直角坐標(biāo)方程16參數(shù)方程：17當(dāng)在xOy面上的投影區(qū)域D是圓域時(shí)，用柱面坐標(biāo)計(jì)算三重積分比較方便在“先一后二”的二重積分中需要用極坐標(biāo)積分時(shí)，我們實(shí)際上就在使用柱面坐標(biāo)計(jì)算三重積分18例4.4是一個(gè)圓錐體它的投影區(qū)域是一個(gè)圓域宜用柱面坐標(biāo)計(jì)算19例4.5交線的投影柱面投影區(qū)域：20

with(plots):zuimian:=implicitplot3d(z=sqrt(x^2+y^2),x=-2..2,y=-2..2,z=0..1.6,color=black,grid=[20,20,20]):paowumian:=implicitplot3d(z=x^2+y^2,x=-2..2,y=-2..2,z=0..1.8,color=red,grid=[20,20,20]):x_axis:=plot3d([u,0,0],u=-1.2..1.2,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=-1.2..1.2,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=-0.2..2,v=0..0.01,thickness=2):display(zuimian,paowumian,x_axis,y_axis,z_axis,orientation=[23,66],scaling=constrained);21例4.57.4三重積分（柱、球）22作的剖面圖：上邊界曲面：下邊界曲面：23題1(3)在xOy面上的投影區(qū)域：24下邊界曲面：上邊界曲面：錐面球面25球面坐標(biāo)Sphericalcoordinates球面坐標(biāo)26球面坐標(biāo)的三組坐標(biāo)面常數(shù)球面：常數(shù)球面坐標(biāo)因此得名27常數(shù)半平面28常數(shù)錐面：29球面坐標(biāo)的坐標(biāo)面30球面坐標(biāo)的坐標(biāo)面動(dòng)畫runinMaplewith(plots):x_axis:=plot3d([u,0,0],u=0..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3.3,v=0..0.01,thickness=2):forifrom0to40dozuimian[i]:=plot3d([r,theta,Pi*i/40],r=0..3,theta=0..2*Pi,color=green,style=patch,coords=spherical):qiumian[i]:=plot3d([2*i/40,theta,phi],theta=0..2*Pi,phi=0..Pi,style=patch,coords=spherical):banpingmian[i]:=plot3d([r,2*Pi*i/40,z],r=0..3,z=-3..3,color=brown,style=patchnogrid,coords=cylindrical):od:banpingmian:=display(seq(banpingmian[i],i=1..40),insequence=true):qiumian:=display(seq(qiumian[i],i=1..40),insequence=true):zuimian:=display(seq(zuimian[i],i=1..40),insequence=true):display(qiumian,banpingmian,zuimian,x_axis,y_axis,z_axis,orientation=[24,67],scaling=constrained);31with(plots):x_axis:=plot3d([u,0,0],u=0..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3.3,v=0..0.01,thickness=2):banpingmian1:=plot3d([r,Pi/4,z],r=0..3,z=0..2,coords=cylindrical,style=patch,color=green):banpingmian2:=plot3d([r,Pi/2.5,z],r=0..3,z=0..2,coords=cylindrical,style=patch,color=grey):zuimian1:=plot3d([r,theta,Pi/4],theta=0..2*Pi,r=0..2,coords=spherical,style=patch,color=red):zuimian2:=plot3d([r,theta,Pi/3],theta=0..2*Pi,r=0..3,coords=spherical,style=patch,color=yellow):qiumian1:=plot3d([1,theta,phi],phi=0..Pi,theta=0..2*Pi,coords=spherical,style=patch,color=brown):qiumian2:=plot3d([1.5,theta,phi],phi=0..Pi,theta=0..2*Pi,coords=spherical,style=patch,color=grey):display(banpingmian1,banpingmian2,zuimian1,zuimian2,qiumian2,x_axis,y_axis,z_axis,orientation=[5,60]);32with(plots):x_axis:=plot3d([u,0,0],u=0..1.5,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..1.5,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..1.5,v=0..0.01,thickness=2):banpingmian1:=plot3d([r,Pi/5,phi],r=1..1.5,phi=Pi/5..Pi/3,coords=spherical,style=patch,color=green):banpingmian2:=plot3d([r,Pi/3,phi],r=1..1.5,phi=Pi/5..Pi/3,coords=spherical,style=patch,color=green):zuimian1:=plot3d([r,theta,Pi/5],theta=Pi/5..Pi/3,r=1..1.5,coords=spherical,style=patch,color=yellow):zuimian2:=plot3d([r,theta,Pi/3],theta=Pi/5..Pi/3,r=1..1.5,coords=spherical,style=patch,color=yellow):qiumian1:=plot3d([1,theta,phi],phi=Pi/5..Pi/3,theta=Pi/5..Pi/3,coords=spherical,style=patch,color=red):qiumian2:=plot3d([1.5,theta,phi],phi=Pi/5..Pi/3,theta=Pi/5..Pi/3,coords=spherical,style=patch,color=red):display(banpingmian1,banpingmian2,zuimian1,zuimian2,qiumian1,qiumian2,x_axis,y_axis,z_axis,orientation=[5,74]);33with(plots):x_axis:=plot3d([u,0,0],u=0..1.5,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..1.5,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..1.5,v=0..0.01,thickness=2):banpingmian1:=plot3d([r,Pi/5,phi],r=1..1.5,phi=Pi/5..Pi/3,coords=spherical,style=patch,color=green):banpingmian3:=plot3d([r,Pi/5,phi],r=0..1.6,phi=Pi/7..Pi/2.5,coords=spherical,style=wireframe,color=green):banpingmian2:=plot3d([r,Pi/3,phi],r=1..1.5,phi=Pi/5..Pi/3,coords=spherical,style=patch,color=grey):banpingmian4:=plot3d([r,Pi/3,phi],r=0..1.6,phi=Pi/7..Pi/2.5,coords=spherical,style=wireframe,color=green):zuimian1:=plot3d([r,theta,Pi/5],theta=Pi/5..Pi/3,r=1..1.5,coords=spherical,style=patch,color=yellow):zuimian3:=plot3d([r,theta,Pi/5],theta=Pi/7..Pi/2.5,r=0..1.8,coords=spherical,style=wireframe,color=blue):zuimian2:=plot3d([r,theta,Pi/3],theta=Pi/5..Pi/3,r=1..1.5,coords=spherical,style=patch,color=yellow):zuimian4:=plot3d([r,theta,Pi/3],theta=Pi/7..Pi/2.5,r=0..1.8,coords=spherical,style=wireframe,color=blue):qiumian1:=plot3d([1,theta,phi],phi=Pi/5..Pi/3,theta=Pi/5..Pi/3,coords=spherical,style=patch,color=red):qiumian3:=plot3d([1,theta,phi],phi=Pi/7..Pi/2.5,theta=Pi/7..Pi/2.5,coords=spherical,style=wireframe,color=red):qiumian2:=plot3d([1.5,theta,phi],phi=Pi/5..Pi/3,theta=Pi/5..Pi/3,coords=spherical,style=patch,color=red):qiumian4:=plot3d([1.5,theta,phi],phi=Pi/7..Pi/2.5,theta=Pi/7..Pi/2.5,coords=spherical,style=wireframe,color=red):display(banpingmian1,banpingmian2,zuimian1,zuimian2,qiumian1,qiumian2,banpingmian3,banpingmian4,zuimian3,zuimian4,qiumian3,qiumian4,x_axis,y_axis,z_axis,orientation=[5,74]);34用以上三組坐標(biāo)面劃分區(qū)域兩張相鄰的球面面兩個(gè)相鄰的圓錐面兩張相鄰的半平面一個(gè)典型的小區(qū)域如圖其體積約為35球面坐標(biāo)下的體積元素：36三重積分從直角坐標(biāo)變換為球面坐標(biāo)37常見曲面的球面坐標(biāo)方程直角坐標(biāo)方程球面坐標(biāo)方程球面with(plots):x_axis:=plot3d([u,0,0],u=0..2,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..2,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..2,v=0..0.01,thickness=2):zuobiaoxi:=display(x_axis,y_axis,z_axis):qumian:=plot3d([sin(u)*cos(v),sin(u)*sin(v),cos(u)],u=0..Pi,v=0..2*Pi,numpoints=800):display(qumian,zuobiaoxi,orientation=[35,60]);38直角坐標(biāo)方程球面坐標(biāo)方程球面例4.639直角坐標(biāo)方程球面坐標(biāo)方程正圓