Image:Archimedes Parabola.png

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[edit] Summary

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

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

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

The sum of this geometric series is 4/3.

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	Date/Time	Dimensions	User	Comment
current	00:08, 10 August 2007	500×437 (19 KB)	Jim.belk (Talk \| contribs)	(Archimedian 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, )