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Another Double Integral

  1. Jul 4, 2013 #1

    FeDeX_LaTeX

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    1. The problem statement, all variables and given/known data

    Find the volume of the region common to the intersecting cylinders ##x^2 + y^2 = a^2## and ##x^2 + z^2 = a^2##.

    3. The attempt at a solution

    I am totally stuck here. What do they mean when they say 'intersecting cylinders'? I've drawn graphs of circles of radius a, centred at the origin, in the x-y plane and the x-z plane. I've put them together and ended up with two identical circles cutting each other at right angles, and I don't see any cylinders... can anyone help me visualise this?

    They have ended up with

    [tex]8 \int_{x=0}^{a} \int_{y=0}^{\sqrt{a^2 - x^2}} z dy dx[/tex]

    I can understand where the limits of integration come from, but not the factor of 8, nor what is actually going on here...
     
  2. jcsd
  3. Jul 4, 2013 #2

    LCKurtz

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  4. Jul 4, 2013 #3

    FeDeX_LaTeX

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    Thanks -- a picture really helped. I found it impossible to visualise.
     
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