Gabriel's Horn is famous for having an infinite surface area but a finite volume. Prove that the volume is finite.

We are asked to confirm that the volume of the figure known as Gabriel's Horn is finite.
We will use the fact that int_1^( oo) (dx)/x^p={[[1/(p-1),"if" p>1],["diverges", p<=1]]
The solid is generated by revolving the unbounded region between the graph of f(x)=1/x and the x-axis, about the x-axis for x>=1 .
We use the disk method: each disk is a circle of radius f(x).
V=pi int_1^(oo) (1/x)^2dx
V=pi int_1^(oo) (dx)/(x^2)
Using the Lemma above we get:
V=pi(1/(2-1))=pi which of course is finite.
http://mathworld.wolfram.com/GabrielsHorn.html


Gabriel's Horn is made by revolving the function f(x)=1/x about the x-axis, with the domain 1lt= x . The volume can be found by slicing the horn up into infinitesimal circles of radius f(x) . Then summing their areas from 1 to oo .
 
V=int_1^oo pi r^2 dx=pi int_1^oo f(x)^2 dx
V=pi int_1^oo 1/x^2 dx=pi(-1/x)|_1^oo=pi [(lim_(x-gtoo)-1/x)+1/1]
V=pi(0+1)
V=pi
As you can see the volume of the horn is exactly pi .

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