File:Archimedes Parabola.png

English: Archimedean decomposition of a parabolic segment into infinitely many triangles. Each green triangle is 1/8 of the area of the blue triangle, each yellow triangle is 1/8 the area of a green triangle, and so forth.

Assuming the blue triangle has area 1, the total area under the parabola is

${\displaystyle 1\,+\,{\frac {2}{8}}\,+\,{\frac {4}{64}}\,+\,\cdots }$

${\displaystyle =1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,\cdots }$.

The sum of this geometric series is 4/3.