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Homework Help: Triple integral depending on a parameter

  1. Feb 4, 2013 #1
    1. The problem statement, all variables and given/known data

    Find [tex] \iiint (x^{2n} + y^{2n} + z^{2n})\,dV [/tex] where the integral is taken over the region of 3D space where [tex] x^{2} + y^{2} + z^{2} \leq 1 [/tex]

    2. Relevant equations

    3. The attempt at a solution

    I tried doing this in Cartesian coordinates, but the limits of integration got very messy and I got stuck after doing the first integral. I also tried using spherical polar coordinates, and then the limits of integration are quite simple, but the integrand gets complicated, unless [itex] n = 1 [/itex], in which case the integral is quite easy to do.

    I then thought that, since the only case where this looks simple enough to do directly is [itex] n = 1 [/itex], I could try to make a conjecture as to what the value of the integral is for general n and then try to prove it by induction. The problem with that, though, is that I don't see how to go from the case [itex] n = k + 1 [/itex] to the case [itex] n = k [/itex].

    Any ideas?
  2. jcsd
  3. Feb 4, 2013 #2


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    I haven't tried it myself yet, and the wife wants the computer just now, but have you tried spherical coordinates? That would be the natural first choice. May or may not work...

    [Edit later] I think you can work it with spherical coordinates using symmetry and the formula$$
    \int_0^\frac \pi 2 \sin^n x\, dx =\int_0^\frac \pi 2 \cos^n x\, dx =
    \frac{1\cdot 3\cdot 5\cdot\cdot\cdot(n-1)}{2\cdot 4\cdot 6\cdot\cdot\cdot n}\frac \pi 2 $$which is valid for ##n## even and ##n \ge 2##.
    Last edited: Feb 4, 2013
  4. Feb 4, 2013 #3


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    This integral covers the volume enclosed by a sphere with radius = 1. It can be broken up into the sum of its parts, since the integrand is a sum.

    For n = 1, the integral will evaluate to the sum of the inertias of a sphere about the three coordinate axes, assuming a unit density for the sphere.
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