Volumes by Cylindrical Shells

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Volumes by Cylindrical Shells
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• What is the volume of and
  y0 revolved around about the y-axis ?
• - since its revolving about the y-axis, the
  equation needs to be solved in terms of y
• - not easy to solve the equation in terms of y
  

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Another Method

• Cylindrical shells can be used
• Cut into cylindrical sections and solve for the
  volume of the cylinder
• Volume circumference x height x thickness

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• Radius is the x position
• ?r is ?x or dx
• The h is the y position or f(x)

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(No Transcript)
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Example

• Find the volume of the solid obtained by rotating
  the region bounded by
• and y0 about the y-axis

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Example

• Find the volume of the solid obtained by rotating
  the region bounded by y x and y x2 about the
  y-axis