Area under a Parabola by Symmetries

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1.
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The area
A
of the region

under the curve
y=
2
x
over the interval
[0,1]
equals the area of the region
ℬ
(in light blue) under the curve and above the line
y=x-1/2
. The area-preserving shear-translation symmetry
(x,y)↦x-
1
2
,y-x+
1
4
of the curve moves the region
ℬ
to a region whose area is one-quarter plus twice the area of the region

under the curve over the interval
[0,1/2]
. The scaling symmetry
(x,y)↦
1
2
x,
1
4
y
​of the curve maps

to

and reduces the area by the factor
1/8
. Thus
A=
1
4
+2×
1
8
A
so
A=1/3
.

Details

Like any parabola, the parabola
y=
2
x
admits two one-parameter families of symmetries. One family consists of the scale-scale transformations
(x,y)↦σx,
2
σ
y
, where
σ≠0
, which scales areas by the factor
3
σ
. The other family consists of the "shear-translation" symmetries given by
(x,y)↦x+t,y+2xt+
2
t

, where
t
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
1/3
. 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

Gerry Harnett
​
​"Area under a Parabola by Symmetries"​
​http://demonstrations.wolfram.com/AreaUnderAParabolaBySymmetries/​
​Wolfram Demonstrations Project​
​Published: February 1, 2013