9478

Logarithmic Property of the Quadrature of the Hyperbola

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.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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,
.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+