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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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HW:7.2#115,1927,35,63(calc)

PracticeinProblemSetup

StepbyStepProcedure

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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Fromgeometry,wefindvolumesofreadilydefinedgeometricfigures.Forexample:

GeometricFigure

Sphere

RightCircularCone

RightCircularCylinder

Volume

V=4r33

V=1r2h3

V=r2h

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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Focuson:

RightCircularCylinder

Volume

V=r2h

Considerasodacan,andflipitontoit'sside.

Tobegin:

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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y=3

thickness:

x=2.01.9=0.1

canbedifferentcurves

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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Fornonstandardgeometricalfigures,thereisnoformula.Instead,usecalculus.

Knowngeometricalfigurescanuseaformula,orcalculus.

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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comparetoformulaforcone

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

HomeworkPart1

7.2 VolumesofRevolution:theDiskMethod

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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start

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDisk/WasherMethod

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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finish

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

HomeworkPart1

7.2 VolumesofRevolution:theDiskMethod

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

HomeworkPart1

7.2 VolumesofRevolution:theDiskMethod

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Rotatearoundtheyaxis(oraverticalline)

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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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.

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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Findthevolumeofthesolidsgeneratedbyrevolvingtheregionboundedby:

y=2x2y=0x=2

aboutthegivenaxes.

a)yaxisb)xaxisc)y=8d)x=2

1st)sketch

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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a)yaxisb)xaxisc)y=8d)x=2verticalhorizontalhorizontalvertical

2nd)axisofrev

3rd)ref.rectangle:_|_toaxisrev

link link link link

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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a)yaxisvertical

2nd)axisofrev

3rd)ref.rectangle:_|_toaxisrev

4th)decide:dyordx?

y=dy

y=dy

( )dycd

5th)decide:R2orR2r2?

ref.rectangledoesNOTtouchtheaxisofrevol.UseR2r2

returntoproblem

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

HomeworkPart1

7.2 VolumesofRevolution:theDiskMethod

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b)xaxishorizontal

2nd)axisofrev

3rd)ref.rectangle:_|_toaxisrev

4th)decide:dyordx?

x=dx

x=dx

( )dxab

5th)decide:R2orR2r2?

ref.rectangletouchestheaxisofrevol.UseR2

returntoproblem

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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c)y=8horizontal

2nd)axisofrev

3rd)ref.rectangle:_|_toaxisrev

4th)decide:dyordx?

x=dx

x=dx

( )dxab

5th)decide:R2orR2r2?

ref.rectangledoesNOTtouchtheaxisofrevol.UseR2r2

returntoproblem

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

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7.2 VolumesofRevolution:theDiskMethod

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d)x=2vertical

2nd)axisofrev

3rd)ref.rectangle:_|_toaxisrev

4th)decide:dyordx?

y=dy

y=dy

( )dycd

5th)decide:R2orR2r2?

ref.rectangledoestouchestheaxisofrevol.UseR2

returntoproblem

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

HomeworkPart1

7.2 VolumesofRevolution:theDiskMethod

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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

HomeworkPart1

7.2 VolumesofRevolution:theDiskMethod

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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