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

When you revolve the region between two curves around an axis instead of the region under one curve, the solid you get is not solid all the way through. There is a gap in the middle, because the inner curve stays away from the axis. Slice that solid perpendicular to the axis and each cross section looks like a washer: a disc with a smaller disc cut out of its center. To get its area you need two radii, one for the outer edge traced by the farther curve and one for the inner edge traced by the closer curve.

The outer radius R and inner radius r each depend on x (or y). The area of one washer slice is π R² minus π r², the big disc's area minus the hole's area. Integrate that area function across the interval where the two curves bound the region, and you get the volume: integral of π(R² - r²) dx. Everything else works exactly like the disc method, you are just subtracting one integral of π r² from another of π R², or combining them into a single integral before evaluating.

This sits right next to the disc method, which is just the washer method with r = 0, and right before shell method problems, which handle solids that are awkward to slice perpendicular to the axis. Setting up R and r correctly, matching each one to the right curve and making sure you know which curve is farther from the axis across the whole interval, is most of the work. The algebra and the Fundamental Theorem handle the rest.