Chapter 7.8, Problem 55ES

Use the results in Exercise 52.

Let R be the region to the right of $x=1$ that is bounded by the $x-\text{axis}$ and the curve $y=1/x.$ When this region is revolved about the $x-\text{axis}$ it generates a solid whose surface is known as Gabriel’s Horn (for reasons that should be clear from the accompanying figure). Show that the solid has a finite volume but its surface has an infinite area. [Note: It has been suggested that if one could saturate the interior of the solid with paint and allow it to keep through to the surface, then one could paint an infinite surface with a finite amount of paint! What do you think?]