Gabriel's Horn, or sometimes known as Torricelli's Trumpet, is the name given to a surface of revolution that has some puzzling properties. The surface resembles a horn, open at both ends, hence the name and has the strange property that we can fill it with paint, but that amount of paint is not enough to paint the inside of the surface...

The Problem

Let's construct Gabriel's Horn and then you can try to prove that it has a finite volume (and to calculate it) and that it has an infinite surface area.

Let's start with ??f(x) = \frac{1}{x}?? between ??x = 1?? and ??x = \infty??.

Now imagine that this is rotated through 360° around the ??x?? axis. It would make a shape that resembles a horn, open at both ends, as shown in the figure below (it is the surface area of this shape that we wish to find later):

Finally, consider the area between the function ??f(x)?? and the ??x?? axis, as shown in the figure below, if this were rotated it would produce a 'filled' horn, which has the volume that we are interested in.

Can you find expressions for the volume and surface area of Gabriel's Horn and evaluate the volume and show that the surface area is infinite?