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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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
Thanks.#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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#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
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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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#10 2013-03-04 17:35:50
- anonimnystefy
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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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón