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

of 16/16
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)

Post on 10-May-2018

222 views

Category:

Documents

Embed Size (px)

TRANSCRIPT

• 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

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

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

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

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

or

Volumeoftheshell=volumeoftheoutercylindervolumeoftheinnercylinder.

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