The Washer Method

Calculus 2 · Axiom Academy

Volumes of revolution when the solid has a hollow center — the disk method, generalized to an outer and an inner radius. 1. The Problem: A Region with a Gap Consider the region between two curves — say (the upper curve) and y=x^2 (the lower curve) — from x=0 to x=1 . Rotate that region about the x -axis and the resulting solid has a hollow center : it isn't completely filled, because the lower curve sweeps out a tunnel the upper curve never reaches. A washer is a disk with a hole in the middle — exactly the metal washer you'd slip onto a bolt. Each cross-section perpendicular to the axis of rotation is one of these, with two radii read straight off the curves at that slice: Distance from the axis up to the outer curve. Here . Distance from the axis up to the inner curve. Here r(x)=x^2 . The area of one washer is the area of the outer circle minus the area of the inner circle: To get the total volume, give each washer a thin width dx , so its volume is , and add them up along the axis of rotation. Summing infinitely many infinitely thin washers is exactly an integral: 4. Worked Example: Between and y=x^2 Find the volume when the region between (upper) and y=x^2 (lower), from x=0 to x=1 , is rotated about the x -axis. The animation sweeps the slice from 0 to 1 and accumulates the running volume; the readout settles on the exact answer. Step 1 — Identify the radii at position x . Outer radius: (up to the upper curve). Inner radius: r(x)=x^2 (up to the lower curve).

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