integration of generalized function

Colin Cowan

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Jul 17, 2004, 12:54:47 AM7/17/04

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I was looking through one of the Calc books I own, and found this
problem:

Show that, for some f continuous on [0,b],

integrate[ f(x)/( f(x) + f(b-x) ) , x , 0 , b]=b/2

The problem was introduced early in the discussion of integrals, and I
have been unable to find complementary problems in other texts.
Various substitutions have failed, as well as applications of The
Fundamental Theorem because F(b), F(0) remain ambiguous.
Simplification and symbolic integration with webMathematica has also
failed because f(x) is evaluated as f*x (which interestingly gives
integrate(x/b,x,0,b)=x^2/(2b)+C ) and F[x_] generates an error. Has
anyone else come across a similar problem to this one?

Klueless

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Jul 17, 2004, 1:17:23 AM7/17/04

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"Colin Cowan" <colinth...@yahoo.com> wrote in message news:d0d07e33.04071...@posting.google.com...

> integrate[ f(x)/( f(x) + f(b-x) ) , x , 0 , b]=b/2

Change of variable y=b-x gives

integrate[ f(x)/( f(x) + f(b-x) ) , x , 0 , b]=integrate[ f(b-y)/( f(b-y) + f(y) ) , y , 0 , b]

Add these two integrals together and the integrand simplifies to 1.
Make conclusions from these observations.

@y.a.h.o.o.c.o.m Kevin O'Neill

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Jul 17, 2004, 1:24:56 AM7/17/04

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"Colin Cowan" <colinth...@yahoo.com> wrote in message news:d0d07e33.04071...@posting.google.com...

=================================================

If transcribed correctly, the problem asks for one function for which the statement is true -- you are not being asked to prove the
statement is true for every continuous function. Try f(x) = K (any nonzero constant).

I hope you find this helpful.

Kevin O'Neill

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Virgil

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Jul 17, 2004, 3:08:12 AM7/17/04

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In article <d0d07e33.04071...@posting.google.com>,
colinth...@yahoo.com (Colin Cowan) wrote:

Since the problem says "Show that, for SOME f continuous on [0,b]"

take f(x) = k <> 0, any non-zero constant function, and the result is

trivially true.

Colin Cowan

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Jul 17, 2004, 12:09:58 PM7/17/04

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I did not include the second part of the problem:

"Use the results to compute the integral with f(x)=sin(x) and b=1"

While the instance of k solves the trivial case, it does not account
for examples like (f(x)=sin(x),b=1) or (f(x)=e^x,b=1)

Jim Burns

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Jul 17, 2004, 12:47:38 PM7/17/04

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See above in thread:

Klueless wrote:
>
> "Colin Cowan" <colinth...@yahoo.com> wrote in message
> news:d0d07e33.04071...@posting.google.com...

> > integrate[ f(x)/( f(x) + f(b-x) ) , x , 0 , b]=b/2
>

> Change of variable y=b-x gives
>

> integrate[ f(x)/( f(x) + f(b-x) ) , x , 0 , b]=

Virgil

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Jul 17, 2004, 1:31:46 PM7/17/04

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In article <40F9582A...@osu.edu>, Jim Burns <burn...@osu.edu>
wrote:

The original will not necessarily work if f(x) + f(b-x) is zero on the
interval at a point where f(x) or f(b-x) is not zero.

For example, let f(x) = 2*x/b - 1. It is linear so certainly continuous,
but f(x) + f(b-x) = 0 everywhere, so the desired integrands are nowhere
defined, much less integrable.

@y.a.h.o.o.c.o.m Kevin O'Neill

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Jul 17, 2004, 3:15:04 PM7/17/04

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Giving a complete problem statement will be helpful to both respondents
and original poster.

Kevin O'Neill

=================================================

"Colin Cowan" <colinth...@yahoo.com> wrote in message news:d0d07e33.04071...@posting.google.com...