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Volume of sphere cut by two parrallel planes

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data

    A sphere of radius R with centre at the origin is cut by two parallel planes at [itex]z=\pm a[/itex], where a<R. Write, in cylindrical coordinates, a triple integral which gives the volume of that part of the sphere between the two planes. Evaluate the volume by first performing the r,θ integrals and the the remaining z integral.

    2. Relevant equations

    [itex] dV=rdrdθdz [/itex]

    3. The attempt at a solution

    The main probelm here is the setting up of my integral, as the answer I am getting is independant of R, which is then clearly wrong.

    My integral runs from:
    [itex]r=\sqrt{R^2-a^2}[/itex] to [itex] r=\sqrt{R^2-z^2} [/itex]
    [itex] θ=0 [/itex] to [itex] θ=2\pi [/itex]
    [itex] z=-a [/itex] to [itex] z=a [/itex]

    I would expect the answer to depend on R, but it keeps cancelling out when I evaluate the r integral. I would be grateful if someone could explain what is wrong with my limits.
     
  2. jcsd
  3. Apr 9, 2012 #2

    Dick

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    I think r should go from 0 to [itex]\sqrt{R^2-z^2}[/itex]. Shouldn't it?
     
  4. Apr 9, 2012 #3
    Well that is certainly true to get the formula for the volume of the whole sphere, but in this case the minimum value that r takes is [itex] \sqrt{R^2-a^2} [/itex]. Your proposal is one that I have considered, but I don't see how it can be justified.
     
  5. Apr 9, 2012 #4

    Dick

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    You already have the z limits from -a to a. Doesn't that take care of the a dependency? You may be visualizing the r coordinate wrong. It's the distance from the z axis to the edge of your solid parallel to the x-y plane.
     
  6. Apr 9, 2012 #5
    Ah yes, just recognised the problem. I was only imagining r as being the distance to the surface from the z axis, neglecting all the interior volume where it can reduce to 0... Thanks!
     
  7. Apr 9, 2012 #6

    Dick

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    Right. Your original limits would be for the volume of a sphere with a cylinder cut out of it. Interesting that the R cancels, isn't it? You might not guess that to be true looking a picture of it.
     
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