CRGreathouse

Science Advisor

Homework Helper

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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.

[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.