Area under a Parabola by Symmetries
Area under a Parabola by Symmetries
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 .
A
y=
2
x
[0,1]
ℬ
y=x-1/2
(x,y)↦x-,y-x+
1
2
1
4
ℬ
[0,1/2]
(x,y)↦x,y
1
2
1
4
1/8
A=+2×A
1
4
1
8
A=1/3