The Shell Method

Calculus 2 · Axiom Academy

Finding volumes of revolution by wrapping thin vertical strips into nested cylindrical shells. Take a thin vertical strip of the region at horizontal distance x from the y -axis. Its top reaches the curve, so its height is exactly f(x) . Now spin the region around the y -axis: the strip sweeps out a hollow cylinder — a cylindrical shell — of radius x and height f(x) . circumference , times height f(x) , times thickness dx — unroll it like a label off a can. 2. Unroll the Shell Into a Slab Why does the formula carry a ? Slit one shell down its side and flatten it out . The curved wall opens into a flat rectangle: its width is the circumference , its height is f(x) , and its thickness is dx . The volume of that thin slab is the product of the three. The wall unrolls to length — the way around a circle of radius x . The shell was as tall as the curve, so the slab is too. The strip's tiny width becomes the slab's depth. 3. Nest the Shells and Add Them Up One shell is just a sliver. To fill the whole solid, nest shells at every radius from x = a to x = b — like the rings of an onion — and add their volumes. As the shells shrink to zero thickness, the sum becomes an integral. f(x) — the height of the shell dx — the thickness of the shell Both methods give the right volume — one is just easier. The Shell Method shines when the rotation is around a vertical axis and your function is already written as y = f(x) . Cuts perpendicular to the axis (horizontal slices).

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