From 2D to 3D

Calculus 2 · Axiom Academy

How architects turn a stack of flat cross-sections into the volume of a curved skyscraper — with integration. You're sizing a tapered tower — 30 m wide at the base, narrowing to 10 m at the top over 100 m of height — and to order concrete you need one number: its volume . Slice it, stack it, decide it. Drag the slice up and down the building. Each horizontal cut is a circle — a 2D cross-section — and its area A(z) = πr² shrinks as the tower tapers. That's the whole idea: a solid is a stack of slices. Give each slice a thickness Δz and it becomes a thin disk of volume πr² · Δz . Add them all up. Drag the slab count and watch the sum climb toward the exact volume — that limit is the integral ∫ A(z) dz. Here's the architect's real question: how much material does the taper save? Drag the top radius — the silhouette and the integral both redraw live — and compare the tower's volume to a full 30 m cylinder of the same height. One move, three uses: slice it , stack it , decide it . Anywhere a 3D shape can be read as a stack of known cross-sections — a CT or MRI scan building organ volume from slices, a fuel tank sized from its profile, a dam from its changing cross-section — the area function A(z) integrated over the span gives the volume.

This is the written version of the interactive lesson above. See the full Calculus 2 course.