I’ve heard of it too. You would need an analytical balance to get accurate measurements weighing a piece of paper. Just cut out the part you want to take an integral of, then cut out a piece of paper with known size (or cut several pieces with different sizes to get more accurate results) and weigh each of them. I guess this used to be cheaper and faster than using computers when computers were big and expensive.
Or maybe just a powder balance. When I was a kid in the 1960s, Dad did a lot of reloading his own ammunition. We kids had fun doing things like weighing our names (weigh a small piece of paper to get the tare, weigh again to get the loaded weight, subtract) and other miniscule things. As I recall, it was accurate to less than a grain (0.065 gram).
One of the things we did was weigh different shapes of paper to calculate area. Start with a sample of a unit area. Cut out a funky shape and weigh it, then do the math.
