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The method of sweeping tangents

Published online by Cambridge University Press:  01 August 2016

Tom M. Apostol  and
Mamikon A. Mnatsakanian
Tom M. Apostol
Affiliation:
Project MATHEMATICS!, California Institute of Technology, Pasadena, CA 91125 USA
Mamikon A. Mnatsakanian
Affiliation:
Project MATHEMATICS!, California Institute of Technology, Pasadena, CA 91125 USA
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Extract

What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1 ? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area can be obtained more easily without calculus, regardless of the bike’s path, using a dynamic visual approach called the method of sweeping tangents that does not require equations for the curves.

Type
Articles
Information
The Mathematical Gazette , Volume 92 , Issue 525 , November 2008 , pp. 396 - 417
Copyright
Copyright © The Mathematical Association 2008

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References

1. Apostol, Tom M. A visual approach to calculus problems, Engineering and Science 58 No. 3 (2000) pp. 22–31. (An online version of this article can be found on the web site http://www.its.caltech.edu/∼mamikon/calculus.html, which also contains animations demonstrating the method of sweeping tangents with applications.)Google Scholar
2. Mnatsakanian, Mamikon A. On the area of a region on a developable surface, Dokladi Armenian Acad. Sci. 73 (2) (1981) pp. 97–101 (Russian); communicated by Academician V. A. Ambartsumian.Google Scholar
3. Apóstol, Tom M. and Mnatsakanian, Mamikon A. Subtangents – an aid to visual calculus, Amer. Math. Monthly 109 (2002) pp. 525–533.Google Scholar
4. Apóstol, Tom M. and Mnatsakanian, Mamikon A. Tangents and subtangents used to calculate areas, Amer. Math. Monthly 109 (2002) pp. 900–908.Google Scholar
5. Apóstol, Tom M. and Mnatsakanian, Mamikon A. Surprising geometric properties of exponential functions, Math Horizons (September 1998) pp. 27–29.Google Scholar
6. Apóstol, Tom M. and Mnatsakanian, Mamikon A. Area and arclength of trochogonal arches, Math Horizons (November 2003) pp. 24–30.Google Scholar