Universal Parabolic Constant

Author

Eric W. Weisstein
March 3, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/UniversalParabolicConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/UniversalParabolicConstant.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Definition

Simplify[(((2 h (Sqrt[(x^2)+4 (h^2)])+(x^2) Log[((2 h+(Sqrt[(x^2)+4 (h^2)]))/x)])/(4 h))/a)/.{h->a,x->2a},a>0]

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Surfaces of Revolution

ParametricPlot[{Exp[y],y},{y,-10,0},PlotStyle->Red]

2π Integrate[Exp[y](Sqrt[1+Exp'[y]^2]),{y,-∞,0}]//FullSimplify

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Plot[Cos[x],{x,-π/2,π/2},PlotStyle->Red]

2π Integrate[Cos[x](Sqrt[1+Cos'[x]^2]),{x,-π/2,π/2}]//FullSimplify

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Average Distance to the Center of a Square

Integrate[(Sqrt[(((x-(1/2)))^2)+(((y-(1/2)))^2)]),{x,0,1},{y,0,1}]

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