# Area under a Parabola by Symmetries

Requires a Wolfram Notebook System

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

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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 .

Contributed by: Gerry Harnett (February 2013)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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.

## Permanent Citation

"Area under a Parabola by Symmetries"

http://demonstrations.wolfram.com/AreaUnderAParabolaBySymmetries/

Wolfram Demonstrations Project

Published: February 1 2013