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#1 2012-08-07 13:26:11

careless25
Real Member
Registered: 2008-07-24
Posts: 555

Integration

Hi,

I just had a calc final and one question stumped me. But i would really love to know the answer. Can anyone help me solve this?
I have to show that:

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#2 2012-08-07 22:50:07

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,377

Re: Integration

hi careless25

I'm getting a sign difference here, and I cannot see why ???

let

then

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2012-08-08 00:49:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Integration

Hi Bob;

I'm getting a sign difference here, and I cannot see why ???

You could have chosen

as the factorization. Now the result follows.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#4 2012-08-08 04:56:48

careless25
Real Member
Registered: 2008-07-24
Posts: 555

Re: Integration

Thanks bob bundy and bobbym.

Following that we were asked to use that to solve

and told to use polar coordinates.

I was confused on how to go from the above function to a function of x and y.

Last edited by careless25 (2012-08-08 04:57:09)

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#5 2012-08-08 06:01:28

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,377

Re: Integration

Had me stuck too.  But I found this:

http://en.wikipedia.org/wiki/Gaussian_integral

which looks like the thing for you.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#6 2012-08-08 06:16:10

careless25
Real Member
Registered: 2008-07-24
Posts: 555

Re: Integration

Hahaha I could not have thought of that on the final exam. Lost 10% right there sad.

Thanks Bob

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#7 2012-08-08 10:34:05

careless25
Real Member
Registered: 2008-07-24
Posts: 555

Re: Integration

Hi Bob bundy,

Can you verify my calculations below(just for my understanding):



now from here convert to polar co-ordinates
EDIT: I am not sure what the integration limits for theta would be...if it is 0 to pi/2, then this works out.

let u = r^2, du = 2rdr

since we squared the original equation, we square root the final answer so

This doesnt agree with what wolfram and wikipedia get down. Any idea where I went wrong?

Thanks

Last edited by careless25 (2012-08-08 11:25:31)

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#8 2012-08-08 11:52:18

careless25
Real Member
Registered: 2008-07-24
Posts: 555

Re: Integration

I have been looking at this and I realized that if we just square the actual integral and dont use the fomula given to us, we get to the correct answer.
This does not work:


but this does:

Last edited by careless25 (2012-08-08 11:53:00)

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#9 2012-08-08 22:31:02

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,377

Re: Integration

hi careless25

This is getting beyond what I know but here's my thoughts:

The limits for theta would appear to be 0 to 2pi (a whole circle's worth ? )

The question asks for limits for x of 0 to infinity.

The Wiki proof goes from - inf to + inf.

So you'd need to half your answer.

That then agrees with the Wolfram answer

http://www.wolframalpha.com/input/?i=in … o+infinity

So I think you have done this correctly.

Did they really think you would dream that up under exam conditions?  It's worth more than 10% if you can .. it's worth a 'first' !

What did other candidates do?

I can only think they were hoping you would have researched this in advance in which case it's an easy 10%

Oh well, it's done now.  Fingers crossed for a good result.  smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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