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Re: integration frustration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91928] Re: integration frustration
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sat, 13 Sep 2008 05:54:15 -0400 (EDT)
  • References: <gadcke$riv$1@smc.vnet.net>

Replacedwings schrieb:
> Dear All,
> 
> I have a question about a particular integral:
> 
>  
>  i[r]= Integrate[(L-z)(f[Sqrt[z^2+r^2]] -f[z]),{z,0,L}]
> 
> 
> Assumptions->f[0]==0, f[r]>0 if r>0.
> 
> Is i[r]>=0 for all r?
> 
No

> Any Ideas?
> 
Yes

> Help!
> 
> TIA,
> 
> Chris
> 

Hi Chris,

here is a simple counter-example:

In[1]:= i[r_] = Integrate[(L - z)*(f[Sqrt[z^2 + r^2]] - f[z]), {z, 0, L}];
In[2]:= f[x_] := Piecewise[{{1, 0 < x < L/10}, {1/10, L/10 <= x <= L}}]
Assuming[L > 0,
  ((Print[#1]; #1) & )[Factor[i[L/2]]] >= 0 // Simplify]


During evaluation of In[2]:= ((-173 + 50*Sqrt[3])*L^2)/1000
Out[3]= False

f being a strictly increasing function is obviously sufficient for
i[r]>=0 for all real r.

Cheers,
Peter


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