Gabriel's Horn (Area and Volume)

Siune

1. The problem statement, all variables and given/known data
To calculate the area and volume of Gabriel's Horn between [ 1, infinity ).
And at the same time prove that, volume closes to finity, while area (or surface ) goes to infinity.


2. Relevant equations
f(x) = 1/x
f´(x)= -1/x^2

Volume = [itex]\pi[/itex] [itex]^{\infty}_{1}[/itex][itex]\int[/itex] ((f(x))^2 )dx


Area = 2[itex]\pi[/itex] [itex]^{\infty}_{1}[/itex][itex]\int[/itex] | f(x) | * [itex]\sqrt{}[/itex](1+(f´(x))^2) dx
3. The attempt at a solution

First page
Second page

I get the Volume done nicely, but the area? I know i could approximate the √(1+(1/x^4)) = √1 and it would solve easily, but what I'm doing wrong in my integral there? If we insert for example the s = 1, we get ln( 1-1 ) which we know ain't allowed.

So I think my integral is totally off but can't figure out how.

Sincerely yours,
Siune

DivisionByZro

You were on the right track, but it gets to be a messy integral. I would just use the comparison test for integrals.

I don't see where the (-1/2) on the second page, first line went. Your integral seems fine except that you're missing a 1/2 on all your terms. The next step would be to sub back in your substitutions and clean up your answer.

Edit: I also don't understand your limits of integration. You're still integrating from 1 to infinity. If I read your writing correctly, you have them set up from sqrt(2) to 1 on the second page, why?