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Volumes by Revolution Rules

Volumes by Revolution Rules

These are the rules to find volume by revolution using integrals.

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katie.helfand@ehschools.org

September 08, 2021
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  1. Volume by Revolution about an Axis Type of Rectangle Method

    General Integral Specific Integral of Volume by Revolution Perpendicular and Attached DISK ! "($%&'())+ Vertical Rectangle ! ,(-(.))/0. 1 2 Horizontal Rectangle ! ,(-(3))/03 0 4 Perpendicular and NOT Attached WASHER ! "[(6(78$ $%&'())+ − (';;8$ $%&'())+] Vertical Rectangle ! ,[(-(.))/ − (=(.))/]0. 1 2 Horizontal Rectangle ! ,[(-(3))/ − (=(3))/]03 0 4 Parallel to Axis CYLINDRICAL SHELL ! 2" ($%&'())(ℎ8'@ℎ7) Vertical Rectangle T – B ! /,(.)(-(.) − =(.))0. 1 2 Horizontal Rectangle R – L ! /,(3)(-(3) − =(3))03 0 4
  2. Volume by Revolution about a line other than the x-axis

    or y-axis **Adjust radius and height if axis is other than x-axis or y-axis WASHER ! ,[(ABCDE E20FBG)/ − (FHHDE E20FBG)/] I = K horizontal Vertical Rectangle ! ,[(-(.) − L)/ − (=(.) − L)/]0. 1 2 M = K vertical Horizontal Rectangle ! ,[(-(3) − L)/ − (=(3) − L)/]03 0 4 CYLINDRICAL SHELL ! /, (E20FBG)(NDF=NC) Vertical Rectangle M = K vertical K ≤ % ! /,(. − L)(-(.) − =(.))0. 1 2 K ≥ Q ! /,(L − .)(-(.) − =(.))0. 1 2 K = 0 and Q ≤ 0 ! /,(−.)(-(.) − =(.))0. 1 2 Horizontal Rectangle I = K horizontal K ≤ S ! /,(3 − L)(-(3) − =(3))03 0 4 K ≥ & ! /,(L − 3)(-(3) − =(3))03 0 4 K = 0 and & ≤ 0 ! /,(−3)(-(3) − =(3))03 0 4
  3. METHOD AXIS OF ROTATION DIRECTION OF THIN RECTANGLE INTEGRATION VARIABLE

    FORMULA FIGURE DISKS (region under a curve) x Perpendicular (⊥) to the x-axis x " #(%(&))()& * + WASHERS (region between two curves) x Perpendicular (⊥) to the x-axis x " #[(%(&))( − (.(&))(])& * + SHELLS (region under a curve) x Parallel (||) to the x-axis y " (#(0)(1(0)))0 ) 2 SHELLS (region between two curves) x Parallel (||) to the x-axis y " (#(0)[1(0) − 3(0)])0 ) 2
  4. METHOD AXIS OF ROTATION DIRECTION OF THIN RECTANGLE INTEGRATION VARIABLE

    FORMULA FIGURE DISKS (region under a curve) y Perpendicular (⊥) to the y-axis y " #(1(0))()0 ) 2 WASHERS (region between two curves) y Perpendicular (⊥) to the y-axis y " #[(1(0))( − (3(0))(])0 ) 2 SHELLS (region under a curve) y Parallel (||) to the y-axis x " (#(&)(%(&)))& * + SHELLS (region between two curves) y Parallel (||) to the y-axis x " (#(&)[%(&) − .(&)])& * +