Volume with the Washer Method: Revolving Around the x- or y-Axis

When you revolve a flat region around an axis, each thin vertical (or horizontal) slice sweeps out a shape as it spins. If the region touches the axis, that slice sweeps a full disc. But if there's a gap between the region and the axis, the slice sweeps a ring instead — a washer — because the gap itself spins around too, carving empty space all the way through the solid. That empty space becomes a hole running through the whole solid, not just a dent on one side.

To get the area of one washer, you cannot just square the difference of the radii. You need the full disc area, π times outer radius squared, then subtract the hole, π times inner radius squared: Area = π(R² - r²). Squaring the difference instead would be a completely different, smaller number — it throws away the cross terms that account for the actual ring shape. Once you have that area as a function of x (or y), you integrate it across the interval where the region lives, between the bounds where the two curves meet or the limits given.

This builds directly on the disc method from the previous topic: a disc is just a washer whose inner radius has shrunk to zero, so Area = π(R² - 0²) = π R^2. Recognizing which one you need comes down to one question: does the region touch the axis of revolution anywhere on the interval? If yes, at least part of the solid is solid discs. If the region stays separated from the axis the entire time, every cross section is a washer, and that's the setup you'll see most often once curves start pairing up to bound a region away from the axis.