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By Tanya Leise

Abstract

University and is captivated by applied mathematics in many contexts, from accelerating cracks to biological clocks. Of particular interest to her recently has been the mathematics of mechanical devices such as astrolabes and planimeters, which are also great fun to collect. When not doing math, she enjoys cycling and sewing stuffed dragons for her daughter. A classic example of Green’s theorem in action is the planimeter, a device that measures the area enclosed by a curve. Most familiar may be the polar planimeter (see Figure 1), for which a nice geometrical explanation can be found in the book by Jennings [4] and a direct constructive proof using Green’s theorem is given by Gatterdam [2]. Other types include the rolling planimeter, which is particularly suited to a vector calculus course for both ease of use and simplicity of proof, and radial planimeters that integrate functions plotted on circular charts (that is, the function is in polar form, r = f(θ)). In this article, we present simple proofs using Green’s theorem for the rolling and polar planimeters, followed by an analysis of how to design radial planimeters that calculate a desired integral, such as that of the square root of a function marked on a circular chart. These proofs are suitable for use in a vector calculus course and avoid the awkward trigonometric and algebraic calculations required by proofs like that in [2]. While the proofs in this article are probably not new (though the author has not seen them elsewhere), they are not readily available, and so these planimeter proofs are presented with the aim of providing calculus instructors a wonderful supplement for their courses. Other planimeter proofs can be found on the web. For example, see [6] for a geometric analysis and [5] for a vector analysis of the polar planimeter, and see [1] for an explanation of the radial planimeter. Both rolling and polar planimeters are available in mechanical and electronic versions for commercial use (a quick web search will reveal several manufacturers). For classroo

Topics: Pole arm
Year: 2008
OAI identifier: oai:CiteSeerX.psu:10.1.1.173.1636
Provided by: CiteSeerX
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