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## #1 2013-03-04 06:19:26

White_Owl
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### 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 (2013-03-04 06:22:07)

## #2 2013-03-04 09:56:49

anonimnystefy
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### Re: Gabriel's Horn

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

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #3 2013-03-04 10:36:25

White_Owl
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### Re: Gabriel's Horn

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

Now I am not sure what to do next?

## #4 2013-03-04 10:44:41

anonimnystefy
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### Re: Gabriel's Horn

Maybe a substitution v=sqrt(u)?

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #5 2013-03-04 10:48:00

bobbym

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### 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 have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

## #6 2013-03-04 11:06:17

White_Owl
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### 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?

## #7 2013-03-04 11:12:23

anonimnystefy
Real Member

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### Re: Gabriel's Horn

I would say that that is divergent, then.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #8 2013-03-04 11:19:11

bobbym

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### 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 have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

## #9 2013-03-04 16:11:13

White_Owl
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### 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?

## #10 2013-03-04 17:35:50

anonimnystefy
Real Member

Online

### 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!

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment