       Re: Help for solving this Integrate[Sqrt[t*(1-t)* (t-z),{t,0,z} ] NEW!!!!

• To: mathgroup at smc.vnet.net
• Subject: [mg112837] Re: Help for solving this Integrate[Sqrt[t*(1-t)* (t-z),{t,0,z} ] NEW!!!!
• From: Curtis Osterhoudt <cfo at lanl.gov>
• Date: Sat, 2 Oct 2010 05:46:44 -0400 (EDT)

```In:= Integrate[Sqrt[t*(1 - z)*(t - z)], {t, 0, z},
Assumptions -> z > 0]

Out= 1/8 \[Pi] Sqrt[-1 + z] z^2

It helped me a great deal to plot both the integral and the answers (especially the integral) vs. t, with various values of z. Then I recognized the integrand can be cast into a slightly different form:

Integrate[Sqrt[-1*((t - z/2)^2 - z^2/4)], {t, 0, z},
Assumptions -> z > 0 && 0 <= t <= z]

The integral gives
(\[Pi] z^2)/8

which is simply the area of half a circle with radius z/2; the circles are obvious if you plot the integrand. They're later scaled by the Sqrt[z-1] factor.

On Friday, October 01, 2010 03:41:39 Hugo wrote:
> Could any body help me to solve the following integral in
> mathematica?
>
> Integrate[sqrt[t * (1-z)*(t-z),{t,0,z}];  t and z are reals; z>0
>
> I did make a mistake posting the wrong equation yesterday, I apologize
> for that.
>
> Any help would be appreciate,
>
> Hugo
>
>

--
==================================
Curtis Osterhoudt
cfo at remove_this.lanl.and_this.gov
==================================

```

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