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