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

*To*: mathgroup at smc.vnet.net*Subject*: [mg67369] Re: Help: ratio of integral of f(x)^2 to square of integral of f(x)*From*: Bill Rowe <readnewsciv at earthlink.net>*Date*: Tue, 20 Jun 2006 02:15:07 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

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 -- To reply via email subtract one hundred and four