9887

Area under a Parabola by Symmetries

The area of the region under the curve over the interval equals the area of the region (in light blue) under the curve and above the line . The area-preserving shear-translation symmetry of the curve moves the region to a region whose area is one-quarter plus twice the area of the region under the curve over the interval . The scaling symmetry
of the curve maps to and reduces the area by the factor . Thus so .

SNAPSHOTS

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

DETAILS

Like any parabola, the parabola admits two one-parameter families of symmetries. One family consists of the scale-scale transformations , where , which scales areas by the factor . The other family consists of the "shear-translation" symmetries given by , where can be any real number; these transformations are area preserving. Applying certain subsets of these symmetries to the region , we can see that the integral in question is exactly . This approach is easily extended to determine the area of a segment of any parabola. It thus provides a calculus-free proof of the quadrature of the parabola.
    • 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.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
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+