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Volumes with Cylindrical Shell Method

  1. Oct 8, 2011 #1
    1. The problem statement, all variables and given/known data
    Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

    y= 4x2, y=-2x+6

    2. Relevant equations


    y= 4x2, y=-2x+6

    These 2 equations meet at x= -3/2 and x=1

    integral from a to b of (2∏x*f(x)) dx

    The answer is 250pi/3.

    3. The attempt at a solution
    Been spending over an hour with this and can't figure out a way to figure this out by shells. I can do this by discs and got the answer right. However, I want to figure this out by shells. First I drew the graph of these 2 equations to find where they meet at. I then did this:

    [itex]\int^{-3/2}_{1}[/itex] 2[itex]\pi[/itex]x((-2x-6)-(4x2)) dx

    I put this in Wolfram and I did not get the intended answer. I work this out by hand and I am getting something very different from Wolfram. Guidance please! Thanks in advance.
     
  2. jcsd
  3. Oct 8, 2011 #2

    SammyS

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    Do do this problem using cylindrical shells, you need to express x as a function of y. The integral will be with respect to y. In other words, your differential has dy in it.

    If the region to be rotated is indeed above y = 4x2 and below y = -2x+6, then you will need to break your integral into to pieces; one piece from y = 0 to 4, the other from y=4 to 9.
     
  4. Oct 8, 2011 #3
    Ha I figured using y from 4 to 9 but I do not see why 0 to 4. You would be integrating the function itself. Then I don't see what you would do with the 2 integrals.
     
  5. Oct 8, 2011 #4

    SammyS

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    Draw the graph.

    x = ±(√y)/2

    So for y = 0 to 4, use (√y)/2 - (-√y)/2

    What did you use for y = 4 to 9 ?
     
  6. Oct 8, 2011 #5
    I didn't use them, I just found the zeros and got 4 through 9 before, but I kind of didn't know what to do with it. Wouldn't the above just be √y by itself? Here is what I would do now

    ∫ from 0 to 4 (2πy√y)dy - ∫ from 4 to 9 (2pi*y*((√y)/2+.5y-3)dy)

    Would this be right? Thanks for your assistance Sammy S.
     
  7. Oct 8, 2011 #6

    SammyS

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    Yes, that's right.
     
  8. Oct 8, 2011 #7
    Sweet! Thanks for your guidance Sammy!
     
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