Volume with the Washer Method: Revolving Around Other Axes

Every washer has an outer radius and an inner radius measured from the axis of rotation outward to each boundary curve. When that axis sits right on the x-axis or y-axis, the radius is just the function value. Move the axis anywhere else and you have to measure distance to that new line instead. For a horizontal axis y = k, the radius to a curve y = f(x) becomes k - f(x) if the axis is above the curve, or f(x) - k if the axis is below it. For a vertical axis x = h, swap the roles: measure horizontally, using h - g(y) or g(y) - h.

This distance-based thinking is exactly why the outer and inner radii can flip. If the axis of rotation sits above both curves in the region, the curve that was originally lower, the one with the smaller function value, ends up farther from the axis and becomes the outer radius. Flip the axis to below the region and the roles swap again: the curve with the larger value is now farther away. The rule is never 'bigger function value wins' — it's always 'farther from the axis wins.'

This topic is the natural next step after washers around the x-axis and y-axis: same formula, V = π times the integral of (outer radius)² minus (inner radius)², but now every radius is a subtraction problem first. It also sets up the next extension, where the axis cuts through the region itself, splitting it into pieces that need separate integrals because the 'outer' and 'inner' labels can swap partway through.