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Sphere-Cylinder Volume

  1. May 28, 2013 #1
    1. The problem statement, all variables and given/known data
    A cylindrical hole of radius R is drilled through the center of a sphere of radius 3. Use the method of cylindrical shells to find the volume of the remaining solid, given that the solid is 6 cm. high.


    2. Relevant equations
    V= ∫2* ∏rh dr (shell method)


    3. The attempt at a solution

    I think this is the form to be used for this problem, but I am probably mistaken.
    2 * ∫2* ∏ * x (R^2 - x^2) dx with a= r and b= R
    4∏ ∫x (R^2 - x^2) dx

    Then I'm stumped on how to do this :/ Thank you for your help.
     
  2. jcsd
  3. May 28, 2013 #2

    LCKurtz

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    You have a = r but there is no r in the problem. Do you mean a = 3? Otherwise your integral looks OK, so just integrate it.

    [Edit] That should be ##\sqrt{R^2-x^2}## in that integrand.
     
    Last edited: May 28, 2013
  4. May 28, 2013 #3
    Oh wow, I see what I did wrong. So my final answer turns out to be

    4/3 * pi (R^2 - 9) ^(3/2)

    Is this correct? Thank you.
     
  5. May 28, 2013 #4

    LCKurtz

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    I didn't notice you left off the square root over the ##R^2-x^2## in your integral formula, but apparently you had it correct on your paper since that looks like the right answer.
     
  6. May 28, 2013 #5
    Great, thank you so much! I feel more confident about how to do these problems now :)
     
  7. May 28, 2013 #6

    haruspex

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    One interesting corollary of this problem is that if you know the length of the cylindrical hole you can determine the volume remaining without knowing any other dimensions.
     
  8. May 29, 2013 #7
    Thank you for your insight on that. I will keep that in mind if I ever come across such a problem again.
     
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