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B How do you find the volume?

  1. Mar 30, 2016 #1
    Find the volume of the solid generated by revolving the region bounded by y = 4x-x^2 and y = 2x about the y-axis. About the line x = 2.

    Since this is rotated about the y-axis, I know I have to manipulate the equation so I can get to x = something. The problem is I cannot change the equation y = 4x-x^2.

    How would you find the volume that is rotated about the y-axis while the equations are y = form instead of x =? If it is not possible, then what do I have to do in this case?
    How would you find the volume if the axis of revolution is not y = 0?
  2. jcsd
  3. Mar 30, 2016 #2


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    You need to express x in terms of y. Use the completing the square technique.
    Change variables. Replace ##x## by ##x'=x-2##.
  4. Apr 2, 2016 #3


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    You can use the "cylinder method" rather that the "disk method" to integrate with respect to x. Imagine a line extending upward from each value of x. That line cuts the "region bounded by [itex]y = 4x-x^2[/itex] and y = 2x", for x between 0 and 2, in a line segment of length [itex](4x- x^2)- 2x= 2x-x^2[/itex]. Rotating around the y-axis that forms a cylinder of radius x and height [itex]2x- x^2[/itex] so area [itex]2\pi x(2x- x^2)= 2\pi (2x^2- x^3)[/itex]. Taking the "thickness" of each cylinder to be "dx", the whole volume is given by [itex]2\pi \int_0^2 2x^2- x^3 dx[/itex].
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