Logarithmic Property of the Quadrature of the Hyperbola

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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


Snapshots


Details

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,

.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send