Cylindrical Shells

Calculus 1 · Axiom Academy

A different way to slice a solid of revolution — peel it into nested shells, like the layers of an onion. To find the volume of a solid of revolution, you chop it into simple pieces you can measure and add them up. Disks and washers slice across the axis. The shell method does the opposite: it peels the solid into thin nested cylinders — and the magic is that each one, unrolled, is just a flat slab. Watch a single cylindrical shell of radius and height peel open and lay flat. Its width becomes the circumference , its height stays , and it is only thick — so its volume is that slab's volume. A cylindrical shell unrolls into a slab: circumference × height × thickness . That is the shell volume element. Stack the shells to fill the solid Rotate the region under (from ) around the y-axis and it sweeps out a solid. Slide the handle to pack it with more and more nested shells: each shell's unrolled volume gets added up, and that Riemann total closes in on the solid's true volume as the gap shrinks to zero. As , the sum of shells becomes the integral — the exact volume. Each strip sweeps into a shell Here is why a shell's radius is and its height is . Drag the focus point across the region: the thin vertical strip standing at position is a distance from the y-axis and reaches up to the curve, so when it spins around the axis it traces a shell of radius and height . Radius from the axis is ; height to the curve is — exactly the pieces the formula needs.

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