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.
<p>Assuming the blue triangle has area 1, the total area under the parabola is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,+\,{\frac {2}{8}}\,+\,{\frac {4}{64}}\,+\,\cdots }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mn>1</mn><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>2</mn><mn>8</mn></mfrac></mrow><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>4</mn><mn>64</mn></mfrac></mrow><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><mo>⋯<!-- ⋯ --></mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle 1\,+\,{\frac {2}{8}}\,+\,{\frac {4}{64}}\,+\,\cdots }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaa996cd8ba01861cdb2fbeef2b5609945bf43e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:19.889ex; height:5.343ex;" alt="{\displaystyle 1\,+\,{\frac {2}{8}}\,+\,{\frac {4}{64}}\,+\,\cdots }"></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,\cdots }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo>=</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>16</mn></mfrac></mrow><mspace width="thinmathspace"></mspace><mo>+</mo><mspace width="thinmathspace"></mspace><mo>⋯<!-- ⋯ --></mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle =1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,\cdots }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bc0edd6755d9c65960e916c9a954a44d5ddf6a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:22.342ex; height:5.176ex;" alt="{\displaystyle =1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,\cdots }"></span>.</dd></dl>
The sum of this <a href="https://en.wikipedia.org/wiki/geometric_series" class="extiw" title="en:geometric series">en:geometric series</a> is 4/3.
<a href="https://en.wikipedia.org/wiki/User:Jim.belk" class="extiw" title="en:User:Jim.belk">en:User:Jim.belk</a>,uploader
Unknown date<div style="display: none;">Unknown date</div>/2009-02-03
<a href="https://en.wikipedia.org/wiki/File:Archimedes_Parabola.png" class="extiw" title="en:File:Archimedes Parabola.png">en:File:Archimedes_Parabola.png</a>