Estimation by noise
Throw dust at a shape you can't measure. Count what lands inside.
A rounded solid turns in a unit box. Forty-eight thousand random points rain through it — the ones that fall inside light hot. Their share of the box is its volume, printed live as the confidence band draws tight.
Some shapes resist the calculus. So we stop integrating and start throwing dice at them.
A solid you can't box in
The form is a rounded box softly fused with a sphere — a smooth intersection with no clean formula for its volume. It exists here only as the density of hot dust that fell inside it.
Noise, uniformly thrown
Points are scattered uniformly through the unit box on the GPU as an instanced cloud. A signed-distance test asks each one a single question: inside, or out?
The ratio is the answer
The fraction that lands inside, times the box volume of eight, is the estimate. Its error falls like one over the square root of the count — so the band tightens, slowly, forever.
Monte-Carlo integration. The same trick that sized the first atomic bombs and prices options today: when a shape is too awkward to integrate, measure it with luck instead. Enough luck, counted honestly, becomes an answer you can trust.