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anonimnystefy
2013-03-04 17:35:50

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!

White_Owl
2013-03-04 16:11:13

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?

bobbym
2013-03-04 11:19:11

It is definitely divergent. The integral does not exist.

anonimnystefy
2013-03-04 11:12:23

I would say that that is divergent, then.

White_Owl
2013-03-04 11:06:17

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?

bobbym
2013-03-04 10:48:00

Hi;

I would have tried the simple numerical method type sub of u = 1 / x in the beginning.

anonimnystefy
2013-03-04 10:44:41

Maybe a substitution v=sqrt(u)?

White_Owl
2013-03-04 10:36:25

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

Now I am not sure what to do next?

anonimnystefy
2013-03-04 09:56:49

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

White_Owl
2013-03-04 06:19:26

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?