Like any parabola, the parabola
admits two one-parameter families of symmetries. One family consists of the scale-scale transformations
, which scales areas by the factor
. The other family consists of the "shear-translation" symmetries given by
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.