# M53 Lec 4.6 Volume (Cylindrical Shells & Other Examples) -...

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Applications of the Definite IntegralVolumes of Solids of Revolutions (Part 2)Mathematics 53Institute of Mathematics (UP Diliman)Lecture 4.6
For today1Volumes of Solids of Revolution: Cylindrical Shell Method2Other Examples: Volumes of Solids of Revolution
Previously...Two methods to find the volume of a solid of revolution:
Previously...Two methods to find the volume of a solid of revolution:1Disk or Washer Method:
Previously...Two methods to find the volume of a solid of revolution:1Disk or Washer Method:Divide the region into rectangles that areperpendicular to the axis of revolution.
Previously...Two methods to find the volume of a solid of revolution:1Disk or Washer Method:Divide the region into rectangles that areperpendicular to the axis of revolution.2Cylindrical Shell Method:
Previously...If only disks are obtained:
Previously...If only disks are obtained:V=Zbaπr2dw
Disk or Washer MethodIf washers are obtained:
Disk or Washer MethodIf washers are obtained:V=Zbaπ(r22-r21)dw
For today1Volumes of Solids of Revolution: Cylindrical Shell Method2Other Examples: Volumes of Solids of Revolution
The Cylindrical Shell MethodCYLINDRICAL SHELL
The Cylindrical Shell MethodVolume of a Cylindrical Shell:V=π(r22-r21)h
The Cylindrical Shell MethodVolume of a Cylindrical Shell:V=π(r22-r21)hWe factorr22-r21
The Cylindrical Shell MethodVolume of a Cylindrical Shell:V=π(r22-r21)hWe factorr22-r21V=π(r2+r1)(r2-r1)h
The Cylindrical Shell MethodVolume of a Cylindrical Shell:V=π(r22-r21)hWe factorr22-r21V=π(r2+r1)(r2-r1)hIf we letr=r1+r22, then
The Cylindrical Shell MethodVolume of a Cylindrical Shell:V=π(r22-r21)hWe factorr22-r21V=π(r2+r1)(r2-r1)hIf we letr=r1+r22, thenr1+r2=2r