极坐标系下绕极轴旋转一周旋转体体积公式推导

174Vol.17,.420018JOURNALOFMATHEMATICSFORTECHNOLOGYAug.2001(,210016)[]P.Guldin,.,.[];;[]O17212[]C[]100724120(2001)0420106203.,axb,0yy(x).oxV=Pbay2(x)dx,(1)y(x).oy.1,,S0AHBP,0rr(H)V.r=r(H),...:.(i)0AHUP,0rRV1(1).,(1),V1=P3(RsinU)2(RcosU)+RcosARcosUP(R2-x2)dx-P3(RsinA)2(RcosA)=P3R3sin2UcosU-P3R3sin2AcosA+PR3(cosA-cosU)-P3R3(cos3A-cos3U).(2)(ii)V1dV1.A,V1U.(2),dV1=2P3R3sinUcos2U-P3R3sin3U+PR3sinU-PR3cos2UsinUdU=2P3R3sinUdU.[]2000210231©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.$V1=dV1+o($U).(4)$V1:[U,U+$U][0,r(U)].(iii)V.dV.2[H,H+$H]á[0,r(H)]$V(2).M=max[H,H+$H]r(H),m=min[H,H+$H]r(H),(4),23Pm3sinH$H+o($H)$V23PM3sinH$H+o($H),23Pm3sinH+o($H)$H$V$H23PM3sinH+o($H)$H.$H0,r(H),lim$H0m=lim$H0M=r(H).dVdH=2P3r3(H)sinH,dV=2P3r3(H)sinHdH.(5)V=2P3BAr3(H)sinHdH.:(P.Guldin)[1].P.GuldinSVS(N,G),Vx=S2PG.(6)2:[H,H+$H][0,r(H)],23r(H),H,S(xq,yl),yl=1SBA23rsinH12r2dH=13SBAr3(H)sinHdH.(7)P.Guldin,V=2PylS=2P3BAr3(H)sinHdH.3:.z,SAHB,0rr(H)z(3)8:AHB,0U2P,0rr(H),V=m8dV=m8r2sinHdrdHdU=2P0dUBAdHr(H)0r2sinHdr=2P3BAr3(H)sinHdH.(3).,(3),.SV.7014:©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.(i)r=a(1+cosH)(0H2P);(ii)(x2+y2)2=a2(x2-y2).(i)V=2P3P0r3(H)sinHdH=2P3P0a3(1+cosH)3sinHdH=83Pa3.(ii)r2=a2cos2H,V=2P3P0r3sinHdH=4Pa33Pö20(cos2H)32sinHdH=Pa342ln1+2-23.Sy=kx,.S:-P2AHBP2,0rr(H)V=2P3BAr3(H)cosHdH.(8),,dV.,S.dV1,.dV..,,.....[][1],.[M].:,1990.[2].[M].:,1998.80117©1995-2004TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.