Spinning into Solids

Calculus 2 · Axiom Academy

Watch a flat region sweep around an axis and become a 3D solid — then see how calculus measures exactly how much it holds. A flat shape, spun, becomes a solid Take a flat region, pin a straight line next to it, and spin the region a full turn around that line. Every point sweeps out a circle, and the circles stack into a smooth three-dimensional object — a solid of revolution . A rectangle gives a cylinder, a right triangle gives a cone, a half-disk gives a sphere. The same trick turns any curve into a solid. Watch the shaded region beneath the curve swing around the x-axis. As it turns, the surface it traces fills in, and the running total is the volume of the solid — the amount of space it encloses. The volume is just the space the region sweeps out as it goes all the way around — and that is what an integral will let us measure. Where the volume comes from: a stack of disks Slice the solid across the axis and each slice is a thin circular disk . Its radius is just the curve's height at that spot, and its volume is π·(radius)² times its thickness. Slide the handle to cut the solid into more and more disks, and watch their stacked volume close in on the true volume of the solid. As Δx → 0 the stacked disks become the exact solid — that limit is the integral V = π∫ (radius)² dx. The curve's height is the radius — and the axis is your choice

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