7.3 volumes of revolution: the shell method - battaly volumes of revolution: the shell method ......

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Calculus Home Page Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 7.3 Volumes of Revolution: the Shell Method Homework Part 2 Title: Homework (1 of 16)

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  • CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

    HomeworkPart1

    7.3 VolumesofRevolution:theShellMethod

    HomeworkPart2

    Title:Homework(1of16)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

    HomeworkPart1

    7.3 VolumesofRevolution:theShellMethod

    HomeworkPart2

    Usealternatemethod: theShellMethod .

    Startwithreferencerectangle,butthistimetheReferenceRectangleisparalleltotheaxisofrevolution.

    Title:alternateapproach(7of16)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

    HomeworkPart1

    7.3 VolumesofRevolution:theShellMethod

    HomeworkPart2

    Usealternatemethod: theShellMethod .

    Startwithreferencerectangle,butthistimetheReferenceRectangleisparalleltotheaxisofrevolution.

    Title:Shellanimation(8of16)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

    shell_method.swf

  • CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

    HomeworkPart1

    7.3 VolumesofRevolution:theShellMethod

    HomeworkPart2

    p

    h

    p=averageradiusofshellh=heightdxordy=thicknessx

    or

    Title:ShellMethod:Formula(9of16)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • 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)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

    HomeworkPart1

    7.3 VolumesofRevolution:theShellMethod

    HomeworkPart2

    Title:example(14of16)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

    HomeworkPart1

    7.3 VolumesofRevolution:theShellMethod

    HomeworkPart2

    Title:example(15of16)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

  • CalculusHomePageClassNotes:Prof.G.Battaly,WestchesterCommunityCollege,NY

    HomeworkPart1

    7.3 VolumesofRevolution:theShellMethod

    HomeworkPart2

    Title:example(16of16)

    http://www.battaly.com/calc/wcccalc1.htmhttp://www.battaly.com/calc/hw/ch6s3a.htmhttp://www.battaly.com/calc/hw/ch6s3b.htm

    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