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Homework Help: Volume by Cylindrical shells

  1. Feb 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Rotate around the y-axis the region above the graph of y=x3 that is bounded by the lines x=1 and y=8

    2. Relevant equations

    dV= (2pix)(y)(dx)

    3. The attempt at a solution

    dV = (2pix)(y)dx
    dV = (2pix)(x^3) dx
    = 2pix^4

    I integrated from y = 1 to y=8 and I get the wrong answer.

    V = 2∏x=1x=2x^4 dx
    V = 2∏[1/5x^5]21

    Can someone please explain to me, how I would set up this equation? Something to do with the first function subtracting the second function which I think is x=1

    Here is a graph: http://www3.wolframalpha.com/Calculate/MSP/MSP15491a07c8557bi4590b00001ibd52eb6gcc3177?MSPStoreType=image/gif&s=21&w=366&h=298&cdf=RangeControl [Broken]
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Feb 14, 2012 #2


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    The problem is with the following:
    dV= (2πx)(y)(dx)
    The height of the cylindrical shell is 8-y, not y.

    So dV = (2πx)(8-y)(dx)
    Last edited by a moderator: May 5, 2017
  4. Feb 14, 2012 #3

    [STRIKE]So dV = (2∏x)(8-x^3)dx
    = 16∏x-2∏x^4 dx

    V= 2∏∫(8x-x^4) dx ?


    Thank you for helping me!
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