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MathGroup Archive 2006

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Re: Help: ratio of integral of f(x)^2 to square of integral of f(x)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67360] Re: Help: ratio of integral of f(x)^2 to square of integral of f(x)
  • From: dh <dh at metrohm.ch>
  • Date: Tue, 20 Jun 2006 02:14:35 -0400 (EDT)
  • References: <e7589k$l5d$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Ronnen,
this is certainly wrong.
Consider b-a small, say dx, then we can approximate the integral from 0 
to dx by: f[0] dx:
(1) (b-a) Integral[ f(x)^2 dx, {x, a, b} ] -> dx^2 f^2[0]
(2) Integral[ f(x) dx, {x, a, b} ] -> dx f[0]
Now, if dx is small enough, the quadratic term is smaller than the 
linear one.

Here is an example: for f[x_] = Sin[x] the ratio is <1 for a< approx 1.4 
and >1 otherwise.
Daniel

ronnen.levinson at gmail.com wrote:
> Hi.
> 
> I'm trying to determine whether the following ratio
> 
> r = (b-a) Integral[ f(x)^2 dx, {x, a, b} ] /
>               Integral[ f(x) dx, {x, a, b} ]
> 
> is always greater than or equal to one for 0 < f(x) <= 1. All values
> all real.
> 
> I've obtained r>=1 for all tested choices of f(x), but seek guidance to
> find the general answer.
> 
> Yours truly,
> 
> Ronnen Levinson.
> 
> P.S. E-mailed CC:s of posted replies appreciated.
> 


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