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What paths have length?

  • Karl Menger
Chapter
  • 68 Downloads

Abstract

In the classical theory, the length of the curve \(y = f(x)(a \leqslant x \leqslant b)\) is determined by computing the integral \(\int\limits_{a}^{b} {\sqrt {{1 + f{{\prime }^{2}}(x)dx}} }\). Geometrically, this means that in determining the length of an arc we really compute the area of a plane domain. The length of the circular arc \(y = \sqrt {{1 - {{x}^{2}}}} (0 \leqslant x \leqslant b)\) is the area of the plane domain \((0 \leqslant x \leqslant b,0 \leqslant y \leqslant 1\sqrt {{1 - {{x}^{2}}}} )\). If the arc happens to be a quarter of a circle, the domain is not even bounded.

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Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Karl Menger
    • 1
  1. 1.ChicagoUSA

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