Integral arising in estimation of discrete series

CRGreathouse
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Main Question or Discussion Point

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.
 

Answers and Replies

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?
 
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]
 
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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