Saturday, February 27, 2010

Integration by weight

Some randomly firing neurons just gave me a flashback to a very unorthodox means of doing calculus: Integration by weight.

Here is what happened: Some years ago I was taking a class in thermodynamics. As part of this class, there was a lab (I don't specifically remember what it was about, but there may have been a Stirling engine involved); and in the course of this lab, a contraption produced a two plots of some quantities relevant to the physics involved on a large sheet of paper.

For whatever reason, it was necessary of us students to integrate this printed plot (effectively to find it's area). So, we spent some time rubbing our heads as to how to do this. We discussed several options, like dividing it into triangles and doing some sort of approximate estimate that way. But all of these methods seemed very time-consuming and tedious. So it dawned upon us: We could weigh it. Said and done, we cut the blob outlined by the plot out of the paper, as well as some reference samples to get an average weight of the paper.

Worked like a charm. Granted, we had access to an analytical balance with pretty high-precision, but still, there is something very appealing about the fact that this works.


  1. I'm lead to understand that this is how complex integrals were in fact performed in my father's day, when slide rules were the normal means of quickly obtaining answers.
    A paper was made that had a very standardized weight, uniform throughout the paper, listed on the package. Users would cut out a graph of the function to be integrated, weighed it, and multiplied by the number on the package to get the answer, almost exactly equal to the actual mathematical integral.
    Of course, this meant that any lab doing this frequently would be left with oddly cut scraps of paper which weren't terribly helpful for anything else now, unless the exact same graph came up again.

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  3. Very interesting. Me and my friend were talking about hand blown Tequila glasses the other day, and, after I argued him into a corner about it, he told me that instead of using calculus to measure the volume of the glass he could just make a mold of the glass. Then he would measure its volume through water displacement.

  4. Eureka!

    Sometimes analog solutions are the best.