Logarithmic Property of the Quadrature of the Hyperbola

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For , the natural logarithm
is the signed area under the curve
between 1 and
. Its essential property, that
, arises from area-preserving symmetries of the curve.
Contributed by: Gerry Harnett (November 2012)
Open content licensed under CC BY-NC-SA
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A transformation of the form is area-preserving: any region is transformed to a region with the same area, as may be confirmed by applying it to a rectangle with sides parallel to the axes. It is also a symmetry of
: any point on the curve is mapped to another point on the curve. Moving the "slide" controller shows the effect of a continuous family of such transformations between the identity and
acting on the region under the curve over the interval
. With the checkbox you can see the effect on a rectangle placed under the curve.
If denotes the area under the curve over an interval
, the area-preserving and symmetry properties of
tell us that
. This is seen by varying "slide" through its range. Thus
.
That is,
.
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