Chapter15TheVolumeofaSolidofRevolutionCircleCylinderCircleConeSolidofrevolution

isasolidgeneratedbyaplaneregionrevolvesaboutafixedline.Thislineiscalledthe

axis.SphereVolumeofSolidsofRevolutionIfthesolidofrevolutionobtainedbyrevolvingtheplaneregionRboundedbyy=f(x)andx=a,x=bandxaxis.What’sthevolumeofit?xyoChoosexasintervalvariable,andchoosesubinterval[x,x+dx],TheelementofVis:Thenthevolumeofthesolidis:ⅠWashersandDisks(1)Example1Findthevolumeofthesolidthatresultswhentheregionbetweenthecurveandthex-axis,fromx=0tox=4,isrevolvedaboutthex-axis.Example1Solution:

Findthevolumeofthesolidthatresultswhentheregionbetweenthecurveandthex-axis,fromx=0tox=4,isrevolvedaboutthex-axis.IfthesolidofrevolutionobtainedbyrevolvingtheplaneregionRboundedabovebyy=f(x)andbelowbyy=g(x)Ontheintervalx=a,x=b.ⅠWashersandDisks(2)IfthesolidofrevolutionobtainedbyrevolvingtheplaneregionRboundedabovebyy=f(x)andbelowbyy=g(x)Ontheintervalx=a,x=b.xyothevolumeofthesolidis:ⅠWashersandDisksExample2Solution:

Findthevolumeofthesolidthatresultswhentheregionboundedbythecurvey=xandy=x2,fromx=0tox=1,isrevolvedaboutthex-axis.0xyVolumeofSolidsofRevolutionIfthesolidofrevolutionobtainedbyrevolvingtheplaneregionRboundedbyx=g(y)andy=c,y=dandyaxis.Chooseyasintervalvariable,andchoosesubinterval[y,y+dy],TheelementofVis:Thenthevolumeofthesolidis:xyocd(3)Example3Findthevolumeofthesolidthatresultswhentheregionboundedbythecurvex=2/y,fromy=1toy=4,isrevolvedaboutthey-axis.Example3Findthevolumeofthesolidthatresultswhentheregionboundedbythecurvex=2/y,fromy=1toy=4,isrevolvedaboutthey-axis.Solution:

IfthesolidofrevolutionobtainedbyrevolvingtheplaneregionRboundedabovebyx=f(y)andbelowbyx=g(y)Ontheintervaly=c,y=d.xyothevolumeofthesolidis:(4)Example4Solution:

Findthevolumeofthesolidthatresultswhentheregionboundedbythecurvex=y2andx=y3,fromy=0toy=1,isrevolvedaboutthey-axis.0xyExample5Solution:

Findthevolumeofthesolidthatresultswhentheregionboundedbythecurvey=x2andy=x,fromx=0tox=1,isrevolvedabouttheliney=-2.0xyy=-2Example6Solution:

Findthevolumeofthesolidthatresultswhentheregionboundedbythecurvey=x2andy=x,fromy=0toy=1,isrevolvedaboutthelinex=-2.0xyx=-2ⅡCylindricalShells0xyIfthesolidofrevolutionobtainedbyrevolvingtheplaneregionRboundedabovebyy=f(x)andbelowbyy=g(x)Ontheintervalx=a,x=b.abⅡCylindricalShells0xyIfthesolidofrevolutionobtainedbyrevolvingtheplaneregionRboundedabovebyx=f(y)andbelowbyx=g(y)Ontheintervaly=c,y=d.dcExample7Solution:

FindthevolumeoftheregionthatwhenBoundedbythecurvethex-ax