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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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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