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Re: integration frustration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91926] Re: [mg91893] integration frustration
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sat, 13 Sep 2008 05:53:53 -0400 (EDT)
  • References: <200809120926.FAA28232@smc.vnet.net>

Replacedwings wrote:
> 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?
> 
> Any Ideas?
> 
> Help!
> 
> TIA,
> 
> Chris

Notice that your i[] is also a function of L.

No, it is not always positive. Best way to show this is to construct a 
counterexample. Look for one that rises and then falls, so that your 
f[Sqrt[z^2+r^2]] will tend to be less than your f[z] (requires picking 
suitable f-dependent r to enforce this).

If this is certifiably not homework, feel free to explain the 
application if you require an explicit counterexample.

Daniel Lichtblau
Wolfram Research




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