Finite area with infinite volume

[tangur]

I know that f(x)= 1/x has a infinite area from 1 to infinity but if this area is roated about the x-axis it gives a finite volume. It's actually Torricelli's trumpet.

However, I can't think of any infinite region with a finite area that when rotated about the y-axis gives an infinite volume.

Any help is welcome.

Thanks

 

[Hurkyl]

In light of that, have you thought about trying to prove that rotating a finite area around the y-axis always yields a finite volume?

 

[tangur]

How would you suggest that I try proving that an infinite region which holds a finite area, always gives a finite volume when rotated around the y-axis.

To be honest it is a question on my assignment, so I'm thinking there has to be an answer for it.

Any hints are accepted