Volumes by Revolution Rules

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

Transcript

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 " (#(&)[%(&) − .(&)])& * +