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Volume of 3 intersecting Cylinders

  1. May 20, 2009 #1
    1. The problem statement, all variables and given/known data

    I am trying to find the volume of three intersecting cylinders that intersect at right angles given that the radius is 5.

    I have found many sites that state just the answer, but I am after the proof for it showing all of the working. I know how to prove the rule for 2 intersecting cylinders, I can't do it for 3 though.


    2. Relevant equations

    The rule I am after is 8(2-sqrt(2))r^3 or any similar form of that.

    3. The attempt at a solution

    V_3(r,r,r) = 16r^3int_0^(pi/4)int_0^1ssqrt(1-s^2cos^2t)dsdt

    A site i have found begins solving with that equation, but I have completely no idea where the numbers come from.
     
  2. jcsd
  3. May 22, 2009 #2

    benorin

    User Avatar
    Homework Helper

    intersection3Cyl.jpg

    The above is the looking-down view of the first octant. The next one is slightly cropped and from the z-axis looking out for getting the bounds of integration with the y=x plane included for reference.

    intersection3Cyl_crop.jpg

    Try setting up the integral now... I'll use cylinders of radius R.

    In cylindrical coordinates [tex](r,\theta, z)[/tex]: Using symmetry (multiply by 16), [tex]0\le \theta\le \scriptstyle{\frac{\pi}{4}}[/tex]
    here, z is bounded above by the red cylinder [tex]x^2+z^2=R^2[/tex] (second image) so [tex]0\le z\le \sqrt{R^2-r^2\cos^2\theta}[/tex]
    and r is bounded by the blue cylinder so [tex]0\le r\le R[/tex]

    so the integral is

    [tex]16\int_{0}^{\scriptstyle{\frac{\pi}{4}}}\int_{0}^{R}\int_{0}^{\sqrt{R^2-r^2\cos^2\theta}}r\, dzdrd\theta = 16\int_{0}^{\scriptstyle{\frac{\pi}{4}}}\int_{0}^{R}r\sqrt{R^2-r^2\cos^2\theta}\, drd\theta[/tex]
    [tex]= -{\scriptstyle{\frac{16}{3}}} R^3\int_{0}^{\scriptstyle{\frac{\pi}{4}}} \frac{\sin^3\theta -1}{\cos^2\theta}\, d\theta = 8R^3\left( 2-\sqrt{2}\right)[/tex]​
     
    Last edited: May 22, 2009
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