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


On 6/19/06 at 12:01 AM, ronnen.levinson at gmail.com wrote:

>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.

No. This ratio is not always greater than or equal to one. Choose f(x) to be x, a = 0 and b = 1, then

In[1]:=
Integrate[x^2, {x,0,1}]/Integrate[x,{x,0,1}]

Out[1]=
2/3

For another example consider f(x) = 1/x. The integral of 1/x from 1 to infinity diverges. But

In[3]:=
Integrate[x^-2, {x,1,Infinity}]

Out[3]=
1

So, there must be a b such that the ratio above for f(x) = 1/x is less than 1
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