Volumes of Solids Swept Tangentially Around Cylinders

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 sweepingtangent 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. 1. FAMILIES OF BRACELETS OF EQUAL VOLUME In Figure 1a, a circular cylindrical hole is drilled through the center of a sphere, leaving a solid we call a bracelet. Figure 1b shows bracelets obtained by drilling cylindrical holes of a given height through spheres of different radii. A classical calculus problem asks to show that all these bracelets have equal volume, which is that of the limiting sphere obtained when the radius of the hole shrinks to zero. It comes as a surprise to learn that the volume of each bracelet depends only on the height of the cylindrical hole and not on its radius or the radius of the drilled sphere! This phenomenon can be explained (and generalized) without resorting to calculus by referring to Figure 2. In Figure 2a, a typical bracelet and the limiting sphere are cut by a horizontal plane parallel to the base of the cylinder. The cross section of the bracelet is a circular annulus swept by a segment of constant length, tangent to the cutting cylinder. The corresponding cross section of the limiting sphere is a circular disk whose radius is easily shown (see Figure 3) to be the length of the tangent segment to the annulus. Thus, each circular disk is a tangent cluster of the annulus which, by Mamikon’s sweeping-tangent theorem, has the same area as the annulus. (See Publication Date: January 27, 2015. Communicating Editor: Paul Yiu. 14 T. M. Apostol and M. A. Mnatsakanian