3D Mass Distribution

Calculus 3 · Axiom Academy

A solid can be denser here than there. To weigh it, chop it into tiny boxes, weigh each one, and add them all up — that sum is a triple integral. How do you weigh something that isn't the same all the way through? A block of the same material everywhere is easy: mass is just density times volume. But real solids — a planet's core, a cast metal part, a loaf of bread — are denser in some places than others . There's no single density to multiply by. The fix is the whole idea of calculus: cut the solid into pieces so small that each one is roughly uniform, weigh each piece, then add. Watch the box get carved into little cells. Each cell lights up in the shade of its own local density — pale where the solid is light, deep blue where it's heavy — and the running total m climbs as the cells add up, one at a time. It settles on the exact mass. The total mass is just accumulated weight — the running sum the boxes leave behind. In the limit of infinitely many infinitesimal boxes, that sum is written . Why "tiny" boxes? Because coarse boxes lie. Estimate the mass by giving every little box a single density — read at one corner — then adding across the box. Drag the slider to slice the solid into more and more pieces along its length. Watch the blocky estimate climb toward the true mass, and the gap shrink toward zero as the boxes get thin. As the slabs vanish and the estimate becomes exact: the discrete sum turns into . That passage to the limit is the triple integral.

This is the written version of the interactive lesson above. See the full Calculus 3 course.