Shell Method

Calculus 1 · Axiom Academy

Find a volume of revolution by adding up nested cylindrical shells — and never invert a function to do it. Take a thin vertical strip at distance x from the axis and spin it around. It sweeps out a hollow cylinder — a shell of radius x , height f(x) , and wall thickness dx . Now slit that shell down the side and flatten it: it becomes a thin rectangular slab . The unrolled wall is a rectangle: width = circumference, height = f(x) One shell holds about . To get the whole solid, sweep the radius from a to b and add every shell — that sum is an integral. Watch it build for y = x^2 rotated about the y -axis on [0, 2] : the running total climbs toward the answer. For y = x^2 on [0, 2] about the y -axis : every shell wraps a full turn around the axis radius = x : distance from the axis out to the shell height = f(x) : how tall the shell stands at that radius dx : the shell's vanishingly small thickness For that same solid, watch the two methods set up side by side. Shells slice parallel to the axis and use f(x) = x^2 directly. Washers slice perpendicular and need the inverse — an algebra detour. When solving for the other variable is ugly, shells win. rotating y = f(x) about the y -axis; solving for x in terms of y is hard; slicing parallel to the axis keeps the function untouched. rotating about the same axis the function is measured from; the other variable solves out cleanly; a perpendicular slice is the natural cut.

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