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

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

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

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

Now I am not sure what to do next?

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Maybe a substitution v=sqrt(u)?

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,302

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

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,302

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

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

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!

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