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Integral arising in estimation of discrete series

  1. Apr 28, 2008 #1

    CRGreathouse

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    I'm trying to solve
    [tex]f(t;a,b)=\int_a^b\sqrt{t-x^3}dx[/tex]
    or find a good estimate for it. The problem is 'nice', and so various niceness assumptions apply: [itex]0\le a\le b\le t[/itex] -- and if other assumptions are needed, they probably hold. :D

    An example of a bad estimate would be [itex](b-a)\sqrt{t-a^3}[/itex] -- I'm looking for a better, or ideally a tractable closed-form version. Mathematica gives a complicated form that seems to be concerned mostly with the case that [itex]x^3>t[/itex] which will not be the case here.

    If it helps, a will be 'close' to 0 and b will be close to t.
     
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  3. Apr 29, 2008 #2

    CRGreathouse

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    Failing a closed solution, any thoughts on a good estimate? Any fans of systems other than Mathematica want to test this integral?
     
  4. Apr 29, 2008 #3

    CRGreathouse

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    Playing around with it a bit, it looks like

    [tex]\int_0^{t^3}\sqrt{t^3-x^3}dx=\frac{\Gamma(4/3)}{\Gamma(11/6)}\sqrt{t^5\pi}[/tex]
     
  5. May 1, 2008 #4

    Gib Z

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    Simpson's rule gives:

    [tex] \int^b_0 \sqrt{b^3 - x^3} dx \approx \frac{b}{6} \left( \sqrt{b^3} + 4\sqrt{\frac{7b^3}{8} \right)[/tex]. Should be reasonable? Which reminds me, I think you meant t instead of t^3 as your upper bound in your last post.
     
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