- #1
Siune
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Homework Statement
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.
Homework 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
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