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Volume Problem

  1. Dec 5, 2007 #1
    1. The problem statement, all variables and given/known data
    What would be the most efficient way to find the volume of the solid x^4+y^4+z^4=1?

    2. Relevant equations

    3. The attempt at a solution

    Cylindrical and spherical coordinates end up messy with integrals that cannot be computed by hand. Im at a loss to find something that will work in the long run! Thanks!
  2. jcsd
  3. Dec 5, 2007 #2


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    Spherical coords. You have to integrate powers of sines and cosines; look these up somewhere. Or integrate by parts. Or use e.g. cos(x)=(1/2)(e^ix + e^-ix) and expand.
  4. Dec 5, 2007 #3
    So i just let x=rcos(o)sin(phi) etc etc? i wouldnt need to try something like x^2=rcos(o)sin(phi)?
    Because i dont see how to find limits for r in the first case...
  5. Dec 5, 2007 #4
    in either case, the jacobian gives me an expression that i can only integrate with mathematica using error functions or elliptical integrals... im starting to think that this is impossible!
  6. Dec 5, 2007 #5
    but there must be a way to do this... no one has any ideas?
  7. Dec 5, 2007 #6


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    Oops, sorry, it's harder than I thought. Cylindrical coords look like your best bet. Do the z integral first (easy), then the rho integral (Mathematica will do it), and finally the phi integral (ditto).

    I got (pi^2 Gamma[1/4])/(3 Gamma[3/4]^3) = 6.48, which is a reasonable number (between a cube of edge length 2 and a sphere of diameter 2).

    EDIT: I also got the same answer with rectangular coords, which is probably even easier.
    Last edited: Dec 5, 2007
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