Hands On History: Area Without Integration: Make Your Own Planimeter

@inproceedings{Foote2007HandsOH,
  title={Hands On History: Area Without Integration: Make Your Own Planimeter},
  author={Robert L. Foote and Edward Sandifer},
  year={2007}
}
Clay tablets from Mesopotamia and papyri from Egypt provide evidence that work with area has been part of mathematics since its early history. These Ancients knew how to find areas of squares, circles, triangles, trapezoids, and a number of other shapes for which we no longer have names. Like many other physical quantities, we usually measure area indirectly. That is, we measure something else, such as lengths, a radius, or angles. Then we do some calculations to find area based on appropriate… 
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Areas of areas generate the shuffle algebra
TLDR
The iterated application of the area operator is shown to be sufficient to recover the iterated-integral signature of a path and all the information added by subsequent levels is equivalent to iterated areas.

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