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Topic: Help for solving this Integrate[Sqrt[t*(1-t)* (t-z),{t,0,z} ] NEW!!!!
Replies: 3   Last Post: Oct 2, 2010 5:46 AM

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Curtis Osterhoudt

Posts: 208
Registered: 6/2/07
Re: Help for solving this Integrate[Sqrt[t*(1-t)* (t-z),{t,0,z} ] NEW!!!!
Posted: Oct 2, 2010 5:46 AM
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In[13]:= Integrate[Sqrt[t*(1 - z)*(t - z)], {t, 0, z},
Assumptions -> z > 0]

Out[13]= 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

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