On the mystery of computing area with a balance

My dustcover-less copy of the 1960 classic, The Scientific American Book of Projects for the Amateur Scientist

Imagine that we have drawn a simple closed curve on a sheet of writing paper. (Simple means that the curve does not cross itself.) This curve marks out an area on our paper. How many ways can you think of to find this area?

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I saw a very clever (and most significant) solution to this problem at a model-airplane meet some years ago. The contest rules required that the models have a fuselage whose cross-sectional area was not less than a certain minimum. This area was easy to check in the good old days when all the models had rectangular cross sections, but with the advent of more streamlined shapes the judges began to have trouble making sure the rules were being followed.

They finally decided to find the required area by first having an accurate drawing of it, and then cutting out the drawing and weighing it. Since the weight per unit area of the paper was known, the area of the cut-out drawing was easy to obtain. Now this solution to the area problem is a splendid example of applied integral calculus. It is a little surprising, then, that people who have actually studied calculus will laugh at the method, or dismiss it as impractical. Yet when an accurate balance or scale is at hand it is the quickest way to determine area.

F. W. Niedenfuhr, Professor of Engineering Mechanics at Ohio State University, from The Pleasures of Mathematics in C. L. Strong's The Scientific American Book of Projects for the Amateur Scientist, 1960. Mr. Strong notes this about Prof. Niedenfuhr's chapter in the table of contents, "The amateur scientist is lured into an encounter with integral calculus." Come for the model airplanes, stay for the integral calculus :-)