Area under a Parabola by Symmetries
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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
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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.
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