Spinning Curves

Calculus 1 · Axiom Academy

What happens when a flat curve becomes a 3D solid? Spin it, slice it, and add up the disks. From a flat curve to a 3D solid A huge family of everyday objects — cans, funnels, vases, light bulbs, even a planet — are solids of revolution : take one flat curve and spin it all the way around an axis. The shape of the curve decides the shape of the solid. The big question this unit answers is simple to ask and surprisingly deep: once you've spun a curve into a solid, how much volume does it hold? Watch the straight line spin around the x -axis. Every point on the line traces a circle, and together they sweep out a smooth cone . The running readout shows the volume filling in as the curve goes all the way around. The curve is the recipe; the solid is what you bake. Spin a different curve and you get a different solid — that's next. Different curves, different solids Pick a curve and watch the solid it sweeps out when it spins around the x -axis. The exact volume of each one is the same integral applied to a different f(x) — you'll see why that formula works in a moment. Same idea every time: each vertical slice of the solid is a circle whose radius is the curve's height, f(x) . Here's how we actually measure the volume. Slice the solid into thin circular disks . Each disk is a flat cylinder: radius r = f(x) , so its face has area , and a tiny thickness makes its volume . Drag the slider to use more, thinner disks and watch the stack close in on the true solid.

This is the written version of the interactive lesson above. See the full Calculus 1 course.