Gabriel's horn is a shape with the paradoxical property that it has infinite surface area, but a finite volume.
Consider the surface area and volume of the solid formed by rotating the region bounded by the -axis, , and , around the -axis. This solid is called Gabriel's horn.
The volume of the solid of revolution can be found using the disk method:
Now consider what happens as we allow to approach infinity:
The surface area of a solid of revolution is given by the formula
In this case, since that gives us
This integral is hard to evaluate, but since in our interval and ,
It follows that:
But , which means that Gabriel's horn has infinite surface area, but a volume of only !