       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)

```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, 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:= i[r_] = Integrate[(L - z)*(f[Sqrt[z^2 + r^2]] - f[z]), {z, 0, L}];
In:= 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:= ((-173 + 50*Sqrt)*L^2)/1000
Out= False

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

Cheers,
Peter

```

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