Math Is Fun Forum

White_Owl

Member

Registered: 2010-03-03

Posts: 106

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)

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Gabriel's Horn

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

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“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
The knowledge of some things as a function of age is a delta function.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

Re: Gabriel's Horn

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

Now I am not sure what to do next?

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Gabriel's Horn

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
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

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?

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

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?

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.