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Ive never seen a source for this, but I had a professor a few years back that a low tech way of calculating the area under a curve (definite integral) was to use a piece of paper with known thickness/density, plot the function, and carefully cut off paper above the line.

Measuring the mass, and then using the density formula, you could calculate the area under the curve.

Was this method used or did I mis-remember the story?

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    $\begingroup$ There's this question from ~8 years ago on Math Stackexchange: math.stackexchange.com/questions/3913/… $\endgroup$ – Robert Furber Apr 11 at 9:04
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    $\begingroup$ This method was still used by chemists 30-40 years year to integrate peaks produced by many instruments and this was pretty popular in chromatography. $\endgroup$ – M. Farooq Apr 11 at 11:17
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    $\begingroup$ He should have used lead sheets -- he'll get more uniform sheet density and need a less precise scale. $\endgroup$ – Carl Witthoft Apr 11 at 12:40
  • $\begingroup$ Why weren't those chemists using a planimeter? en.wikipedia.org/wiki/Planimeter $\endgroup$ – Carl Witthoft Apr 11 at 12:42
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    $\begingroup$ Not sure about planimeters usage, but cut-and-weigh was a standard practice in analytical chemistry (my mentors were proud of this technique). I don't belong to that era of 70s, however, I have used this technique once or twice, just to confirm peak area ratios in chromatography. This was done in order to avoid numerical integration (quick & dirty). Results are pretty reliable. $\endgroup$ – M. Farooq Apr 11 at 14:40
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In fact you do not need to know the thickness of the paper or its density. You only have to be sure that the paper is reasonably uniform (has some constant density). Because you can draw a rectangle next to your curve, and weight it too, and compare with the weight of the area under the curve.

I am not sure who invented this (one can never be sure who did some simple thing first) but Galileo used the method for determining the area under a cycloid, which was not known theoretically at that time.

I do not remember exactly where I read this, but most likely in the book of Drake, Galileo at work.

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