I’ve heard of people printing out charts, then cutting out the part they wanted to calculate an integral of, then weighing the paper.
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.
