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.

Nono that's genius

..ignoring the part that it's just a discrete approximation of an integral a la a Riemann sum.

How thick is your paper?

That's pretty clever!

That was a common way to do it before computers were common.