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单词 EqualArcLengthAndArea
释义

equal arc length and area


We want to determine the nonnegative differentiableMathworldPlanetmathPlanetmath real functions  xy  whose graph has the property that the arc lengthMathworldPlanetmath between any two points of it is the same as the area (http://planetmath.org/AreaOfPlaneRegion) by the curve, the x-axis and the ordinate lines of those points.

The requirement leads to the equation

ax1+(dydx)2𝑑x=axy𝑑x.(1)

By the fundamental theorem of calculusMathworldPlanetmathPlanetmath, we infer from (1) the differential equationMathworldPlanetmath

1+(dydx)2=y,(2)

whence  dydx=y2-1.  In the case  y1,  the separation of variablesMathworldPlanetmath yields

𝑑x=dyy2-1,

i.e.

x+C=arcoshy.

Consequently, the equation (2) has the general solution

y=cosh(x+C)(3)

and the singular solution

y 1.(4)

The functions defined by (3) and (4) are the only satisfying the given requirement.  The graphs are a chain curve (which may be translated in the horizontal direction) and a line parallelMathworldPlanetmathPlanetmath to the x-axis.  Evidently, the line is the envelope of the integral curves given be the general solution.

Titleequal arc length and area
Canonical nameEqualArcLengthAndArea
Date of creation2013-03-22 19:13:36
Last modified on2013-03-22 19:13:36
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id8
Authorpahio (2872)
Entry typeExample
Classificationmsc 53A04
Classificationmsc 34A34
Classificationmsc 34A05
Classificationmsc 26A09
Synonymequal area and arc length
Related topicArcosh
Related topicHyperbolicFunctions
Related topicChainCurve

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