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# Volumes of solids swept tangentially around cylinders

Apostol, Tom M. and Mnatsakanian, Mamikon A. (2015) Volumes of solids swept tangentially around cylinders. Forum Geometricorum, 15 . pp. 13-44. ISSN 1534-1178. https://resolver.caltech.edu/CaltechAUTHORS:20150514-132136696

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## Abstract

In earlier work ([1]-[5]) the authors used the method of sweeping tangents to calculate area and arclength related to certain planar regions. This paper extends the method to determine volumes of solids. Specifically, take a region S in the upper half of the xy plane and allow the plane to sweep tangentially around a general cylinder with the x axis lying on the cylinder. The solid swept by S is called a solid tangent sweep. Its solid tangent cluster is the solid swept by S when the cylinder shrinks to the x axis. Theorem 1: The volume of the solid tangent sweep does not depend on the profile of the cylinder, so it is equal to the volume of the solid tangent cluster. The proof uses Mamikon's sweeping-tangent theorem: The area of a tangent sweep to a plane curve is equal to the area of its tangent cluster, together with a classical slicing principle: Two solids have equal volumes if their horizontal cross sections taken at any height have equal areas. Interesting families of tangentially swept solids of equal volume are constructed by varying the cylinder. For most families in this paper the solid tangent cluster is a classical solid of revolution whose volume is equal to that of each member of the family. We treat forty different examples including familiar solids such as pseudosphere, ellipsoid, paraboloid, hyperboloid, persoids, catenoid, and cardioid and strophoid of revolution, all of whose volumes are obtained with the extended method of sweeping tangents. Part II treats sweeping around more general surfaces.

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Additional Information:© 2015 Florida Atlantic University. Publication Date: January 27, 2015. Communicating Editor: Paul Yiu.
Record Number:CaltechAUTHORS:20150514-132136696
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20150514-132136696
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:57533
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:14 May 2015 21:40