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G.Battaly2011 1
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
HW:7.2#115,1927,35,63(calc)
PracticeinProblemSetup
StepbyStepProcedure
http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s2a.htmhttp://www.battaly.com/calc/hw/ch6s2b.htmhttp://www.battaly.com/calc/hw/ch6s2c.htm
G.Battaly2011 2
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
Fromgeometry,wefindvolumesofreadilydefinedgeometricfigures.Forexample:
GeometricFigure
Sphere
RightCircularCone
RightCircularCylinder
Volume
V=4r33
V=1r2h3
V=r2h
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 3
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
Focuson:
RightCircularCylinder
Volume
V=r2h
Considerasodacan,andflipitontoit'sside.
Tobegin:
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 4
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
y=3
thickness:
x=2.01.9=0.1
canbedifferentcurves
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 5
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 6
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
Fornonstandardgeometricalfigures,thereisnoformula.Instead,usecalculus.
Knowngeometricalfigurescanuseaformula,orcalculus.
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 7
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
comparetoformulaforcone
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 8
G.Battaly2011 9
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 10
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
start
G.Battaly2011 11
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDisk/WasherMethod
HomeworkPart2HomeworkPart3
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 12
http://www.battaly.com/calc/classnotes/
G.Battaly2011 13
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
finish
http://www.battaly.com/calc/classnotes/
G.Battaly2011 14
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
G.Battaly2011 15
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
Rotatearoundtheyaxis(oraverticalline)
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 16
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
G.Battaly2011 17
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
G.Battaly2011 18
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
VolumesofRevolutionDiskMethod1.Sketchthecurvesandidentifytheregion,usingthepointsofintersection.2.Locatetheaxisofrevolutiononthesketch.3.Decidewhethertouseahorizontalorverticalrectangle.Therectangleshouldbeperpendiculartotheaxisofrevolution.4.Sketchtherectangleanddeterminethevariableofintegration. Iftherectangleishorizontal,thenintegratewithrespecttoy(usedy).Theintegrandmustbeintermsofy. Iftherectangleisvertical,thenintegratewithrespecttox(usedx).Theintegrandmustbeintermsofx.5.Determinetheintegrand:R2,orR2r2?a)Iftherectangletouchestheaxisofrevolution,identifyRasthelengthoftherectangle.FindRintermsoftheappropriatevariable(seeabove),anduseR2astheintegrand.b)Iftherectangledoesnottouchtheaxisofrevolution,identifyRasthedistanceofthefurthestendoftherectanglefromtheaxisofrevolutionandrasthedistanceoftheclosestendoftherectanglefromtheaxisofrevolution.UseR2r2astheintegrand.
http://www.battaly.com/calc/calchw.htm#volumes of revolution - disk methodhttp://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 19
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
Findthevolumeofthesolidsgeneratedbyrevolvingtheregionboundedby:
y=2x2y=0x=2aboutthegivenaxes.a)yaxisb)xaxisc)y=8d)x=2
1st)sketch
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 20
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
a)yaxisb)xaxisc)y=8d)x=2verticalhorizontalhorizontalvertical
2nd)axisofrev
3rd)ref.rectangle:_|_toaxisrev
link link link link
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 21
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
a)yaxisvertical
2nd)axisofrev
3rd)ref.rectangle:_|_toaxisrev
4th)decide:dyordx?
y=dyy=dy
( )dycd
5th)decide:R2orR2r2?
ref.rectangledoesNOTtouchtheaxisofrevol.UseR2r2
returntoproblem
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 22
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
b)xaxishorizontal
2nd)axisofrev
3rd)ref.rectangle:_|_toaxisrev
4th)decide:dyordx?
x=dx
x=dx
( )dxab
5th)decide:R2orR2r2?
ref.rectangletouchestheaxisofrevol.UseR2
returntoproblem
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 23
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
c)y=8horizontal
2nd)axisofrev
3rd)ref.rectangle:_|_toaxisrev
4th)decide:dyordx?
x=dx
x=dx
( )dxab
5th)decide:R2orR2r2?
ref.rectangledoesNOTtouchtheaxisofrevol.UseR2r2
returntoproblem
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 24
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
d)x=2vertical
2nd)axisofrev
3rd)ref.rectangle:_|_toaxisrev
4th)decide:dyordx?
y=dyy=dy
( )dycd
5th)decide:R2orR2r2?
ref.rectangledoestouchestheaxisofrevol.UseR2
returntoproblem
http://www.battaly.com/calc/classnotes/http://www.battaly.com/calc/calchw.htm#Chapter 7
G.Battaly2011 25
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
http://www.battaly.com/calc/classnotes/
G.Battaly2011 26
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.2 VolumesofRevolution:theDiskMethod
HomeworkPart2HomeworkPart3
G.Battaly2011 27
G.Battaly2011 28
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Page 1: backgroundPage 2: IntroductionPage 3: Volume of RevolutionPage 4: problemPage 5: need calculusPage 6: examplePage 7: Mar 7-7:32 PMPage 8: examplePage 9: examplePage 10: washer methodPage 11: Mar 9-5:18 PMPage 12: examplePage 13: examplePage 14: revolve about y-axisPage 15: examplePage 16: example about y-axisPage 17: step-by-stepPage 18: practice setupPage 19: example about vertical linePage 20: example about vertical linePage 21: example about y-axisPage 22: example about x-axisPage 23: example about y=8Page 24: example about x=2Page 25: example about horizontal linePage 26: Mar 21-6:54 PMPage 27: Mar 21-6:34 PMPage 28: Mar 21-6:30 PMPage 29: Mar 6-6:46 PMPage 30: Mar 6-6:46 PM