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 Post subject: Posted: Wed, 24 Sep 2008 12:16:50 UTC
 S.O.S. Oldtimer

Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
As commutative said, this is a special case of a general result. Move the inside the integral.
Also, read this thread particularly to commutative's post at the bottom http://www.sosmath.com/CBB/viewtopic.php?t=35977&highlight=

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Re: Calculus (13)Posted: Thu, 25 Sep 2008 00:46:34 UTC
 Senior Member

Joined: Fri, 9 Nov 2007 00:04:38 UTC
Posts: 128
I think I have the actual answer now:
Spoiler:

Let . The kth term of the sum is (n-k) times the area under the curve from to . Summing this over k is equivalent to summing the value of the integral from 0 to .
Or:

Plugging this in gives
Letting and we see that this is really just the riemann sum for

The answer is somewhere around .47
If I'm correct, the general case of this is taking the f(y) to be anything of the form

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"The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people."

Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

Last edited by simplethinker on Fri, 26 Sep 2008 15:30:21 UTC, edited 1 time in total.

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 Post subject: Re: Calculus (13)Posted: Thu, 25 Sep 2008 18:24:00 UTC
 S.O.S. Oldtimer

Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
simplethinker wrote:
Or:

Hi simplethinker, may you please explain this equality again. This is the only line that I do not understand from your solution.

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Re: Calculus (13)Posted: Thu, 25 Sep 2008 21:57:47 UTC
 Senior Member

Joined: Fri, 9 Nov 2007 00:04:38 UTC
Posts: 128
bidentate wrote:
simplethinker wrote:
Or:

Hi simplethinker, may you please explain this equality again. This is the only line that I do not understand from your solution.

Set so
If we set n=4, then the sum is: (4-0)*(g(1) - g(0)) + (4-1)*(g(2) - g(1)) + (4-2)*(g(3) - g(2)) + (4-3)*(g(4) - g(3)) = 4*g(1) - 4*g(0) + 3*g(2) - 3*g(1) + 2*g(3) - 2*g(2) + 1*g(4) - 1*g(3) = (4-3)*g(1) + (3-2)*g(2) + (2-1)*g(3) + (1)*g(4) - 4*g(0) = (g(1)-g(0)) + (g(2)-g(0)) + (g(3)-g(0)) + (g(4)-g(0))
[ for this case g(0)=0, so the above could be simplified by removing the g(0)'s, but I don't think it would generalize well for other functions such that g(0)=0]
Since , we have
And this can be generalized to arbitrary n.

_________________
"The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people."

Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

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 Post subject: Posted: Fri, 26 Sep 2008 14:12:10 UTC
 S.O.S. Oldtimer

Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
Very nice, good job!

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Posted: Fri, 26 Sep 2008 15:29:28 UTC
 Senior Member

Joined: Fri, 9 Nov 2007 00:04:38 UTC
Posts: 128
Thank you
I also just noticed that there was an error in my first explanation of the sum. It should be (n-k) times the area under the curve from y_k to y_k+1.

_________________
"The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people."

Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

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 Post subject: Posted: Fri, 19 Dec 2008 23:06:13 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 18 Mar 2006 08:42:24 UTC
Posts: 834
let's see your solutions to the generalized version of the problem, i.e. prove that:

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 Post subject: Posted: Sat, 20 Dec 2008 06:05:51 UTC
 Senior Member

Joined: Fri, 9 Nov 2007 00:04:38 UTC
Posts: 128
I think my solution just needs a bit of tweaking (and I finally figured out how to switch the order of integration):
Spoiler:
(for the reason see my above post). Set , so the original expression becomes , and remembering the definition of g(y), we have .
This integral is over the region , which can also be expressed as , so finally the expression becomes

Another correction for my solution a couple posts up: I said I didn't think it would generalize well for functions such that g(0)0, but by my definition g(0) has to be identically zero.

_________________
"The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people."

Some people live their lives by the KISS principle (Keep It Simple Stupid), but I prefer a slightly different one:
Make It Fun And Complicate It Further, or MIFACIF. Ever try and find the volume of a cube in spherical coordinates?

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