Chapter 5.6, Problem 46E

Paradox of Gabriel’s horn or the infinite paint can. Through we cannot prove it here, the surface area of Gabriel’s horn (see Exercise 45) is given by

$S={\displaystyle {\int }_1^{\infty }\frac{2\pi }{x}}\sqrt{1+\frac{1}{x^4}}dx$.

Show that the surface area of Gabriel’s horn is infinite. The paradox is that the volume of the horn is finite, but the surface area is infinite. This is like a can of paint that has a finite volume but, when full, does not hold enough paint to paint the outside of the can.

Using the fnlnt feature on a graphing calculator, with successively larger values for the upper limit, you will find that the integral diverges.