Gabriel's Other Possessions

Abstract

Gabriel's Horn is a solid of revolution commonly featured in calculus textbooks as a counter-intuitive example of a solid having finite volume but infinite surface area. Other examples of solids with surprising geometrical finitude relationships have also appeared in the literature. This article cites several intriguing examples (some of fractal type), adds additional ones, and discusses how these topics can enhance a Calculus II or Real Analysis course by adding cohesiveness between topics, challenging students at multiple levels, and illustrating both the power and limitations of a computer algebra system.

Keywords:

• Gabriel's Horn
• solid of revolution
• improper integral
• volume
• surface area

ACKNOWLEDGMENT

The author would like to thank the referees for many helpful comments, especially for suggesting simpler geometrical arguments to replace some algebraic inequalities, and the Venn diagram in Figure 5 to cleanly illustrate all possible relationships.

Notes

¹A referee noted that the seeming paradox is a consequence of our selective use of the properties of physical paint. We insist that paint be able to indefinitely flow through the diminishing interior tube of the Horn, yet disallow the possibility of coating the Horn's exterior with a finite quantity of paint whose thickness decreases exponentially as . The nature and possible resolutions of the paradox were discussed in [3]; we continue to use the colloquial terminology throughout the article.