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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: 15,606

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: 15,606

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: 88,965

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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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: 15,606

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: 88,965

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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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: 15,606

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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