7.3 volumes of revolution: the shell method - battaly volumes of revolution: the shell method ......
TRANSCRIPT
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Title:Homework(1of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Consider:y=x33x+3,x=0,y=0,x=2
Easytorevolveaboutxaxis:Usediskmethod
BUT,whataboutrevolvingaboutyaxis?
Title:Introduction(2of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Consider:y=x33x+3,x=0,y=0,x=2Whataboutrevolvingaboutyaxis?
Referencerectanglefordiskmethodisnotconsistentanddoesnothaveaneasyalgebraicrepresentation.
Title:problemsw.diskmethod(3of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Consider:y=x33x+3,x=0,y=0,x=2
1 2
3
Coulddivideinto3regions.Thenaddthevolumes.
Title:howtosetupdiskmethod(4of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Consider:y=x33x+3,x=0,y=0,x=2
1 2
3
Coulddivideinto3regions.
ForRegion1:
Wehavex,butweneedanintegrandto
matchthedy.
Title:matchvariableinintegrand(5of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Consider:y=x33x+3,x=0,y=0,x=2
1 2
3
Fordy,weneedtosolvethecubicfunctionforxintermsofytoexpresstheintegrandalgebraically
Title:ify=f(x),thenx=?(6of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Usealternatemethod: theShellMethod .
Startwithreferencerectangle,butthistimetheReferenceRectangleisparalleltotheaxisofrevolution.
Title:alternateapproach(7of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Usealternatemethod: theShellMethod .
Startwithreferencerectangle,butthistimetheReferenceRectangleisparalleltotheaxisofrevolution.
Title:Shellanimation(8of16)
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shell_method.swf
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
p
h
p=averageradiusofshellh=heightdxordy=thicknessx
or
Title:ShellMethod:Formula(9of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
p
h
p=averageradiusofshellh=heightdxordy=thicknessx
or
Volumeoftheshell=volumeoftheoutercylindervolumeoftheinnercylinder.
w(deltax)isthewidthofthereferenceshell.Addthevolumesofadjacentshells,andletdeltax>0.Resultsinrepresentationofthethicknessoftheshellasdxordy.
Title:ShellMethod:Why?(10of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 WhentoUsetheShellMethod
HomeworkPart2
VolumesofRevolutionWhichMethod?
1.Sketchthecurvesandidentifytheregion,usingthepointsofintersection.
2.Locatetheaxisofrevolutiononthesketch.
3.Decidewhethertouseahorizontalorverticalrectangle.Selecttheorientationthatrequirestheleastnumberofseparatesections.
4.DecidewhethertousetheDiscMethodortheShellMethod:a)Iftherectangeisperpendicular totheaxisofrevolution,usetheDiscMethod.b)Iftherectangleisparalleltotheaxisofrevolution,usetheShellMethod .
Title:ShellMethod:When?(11of16)
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VolumesofRevolutionShellMethod
1.CompleteSteps1to4inVolumesofRevolution,whichMethod?notedabove.
2.Besurethatyourrectangleisparalleltotheaxisofrevolution.
3.Determinethevariableofintegration:a)Iftherectangleishorizontal,thenintegratewithrespecttoy(usedy).Theintegrandmustbeintermsofy.b)Iftherectangleisvertical,thenintegratewithrespecttox(usedx).Theintegrandmustbeintermsofx.
4.Determinetheintegrand:p(x)h(x)orp(y)h(y)?a)Iftherectangleishorizontal,identifyp(y),thedistanceoftherectanglefromtheaxisofrevolution,andh(y),thelengthoftherectangle.Use:b)Iftherectangleisvertical,identifyp(x),thedistanceoftherectanglefromtheaxisofrevolution,andh(x),thelengthoftherectangle.Use:
CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1 HomeworkPart2
Title:shellmethod:how(12of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Consider:y=x33x+3,x=0,y=0,x=2
Whichmethod?
Whichformula?
ShellMethod.Can'tfindx=f(y).
*
*Usetheformulawithdx!
Title:example(13of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Title:example(14of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Title:example(15of16)
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CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY
HomeworkPart1
7.3 VolumesofRevolution:theShellMethod
HomeworkPart2
Title:example(16of16)
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Page 1: Volumes of Revolution: the Shell MethodPage 2: IntroductionPage 3: problems w. disk methodPage 4: how to set up disk methodPage 5: match variable in integrandPage 6: if y=f(x), then x=?Page 7: alternate approachPage 8: Shell animationPage 9: Shell Method: FormulaPage 10: Shell Method: Why?Page 11: Shell Method: When?Page 12: shell method: howPage 13: examplePage 14: examplePage 15: examplePage 16: example