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#1 2013-03-03 07:19:26

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Gabriel's Horn

According to the textbook, the surface area of the curve y=1/x for x>=1, rotated around x-axis is infinite.
According to my calculations it is finite. I suspect I have a mistake, but I cannot find it. Please help:

Surface area is:


Here we have a=1, b=\infty, and f(x)=1/x
Since one of the bounds is infinity, we have an improper integral and have to do it with a limit:


Looking at the description of Gabriel's Horn in Wikipedia, I see that they used for the surface a function:


Why is that? How did they manage to convert
into 1?

Last edited by White_Owl (2013-03-03 07:22:07)

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#2 2013-03-03 10:56:49

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Gabriel's Horn

Where did you get 4u^6 in the denominator in the step right after the substitution from?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2013-03-03 11:36:25

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Gabriel's Horn


Now I wonder myself where did I got u^6 smile Thanks.


Now I am not sure what to do next?

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#4 2013-03-03 11:44:41

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Gabriel's Horn

Maybe a substitution v=sqrt(u)? dunno


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#5 2013-03-03 11:48:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,362

Re: Gabriel's Horn

Hi;

I would have tried the simple numerical method type sub of u = 1 / x in the beginning.


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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#6 2013-03-03 12:06:17

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Gabriel's Horn

Another attempt:
Starting from here:


Let

Then:

We have a formula #24 in the table of integrals in the textbook:


So:

And here we have first limit is infinity divided by infinity, second limit is infinity and a constant.
Therefore we have an infinity in the final answer...
Did I make any mistakes?

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#7 2013-03-03 12:12:23

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Gabriel's Horn

I would say that that is divergent, then.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#8 2013-03-03 12:19:11

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,362

Re: Gabriel's Horn

It is definitely divergent. The integral does not exist.


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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#9 2013-03-03 17:11:13

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Gabriel's Horn

Divergence should be proven or shown...
I think this solution can be used as a prove, but maybe there is an easier way?

And I repeat the question: Why Wikipedia uses incomplete formula?

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#10 2013-03-03 18:35:50

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Gabriel's Horn

Well, 1/x *sqrt(1+1/x^4) is everywhere greater than 1/x, so its integral on any interval will be greater than the integral of 1/x on the same interval!


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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