Volume with the Disc Method: Revolving Around Other Axes

The disc method always measures volume by stacking thin circular slices, and each slice's radius depends entirely on how far the curve sits from the axis of spinning. When that axis is the x-axis, the radius is just the function value because the x-axis is the zero line. Move the axis anywhere else, and the radius has to count the extra gap between the curve and that new line. Every single radius in the solid changes, so the whole integral changes with it.

To write the new radius, treat the axis as a reference line and measure distance to it like a subtraction problem. If the axis sits below the region, at y = -k, the radius is the function value plus k, since you are stretching down past zero and then further down to the axis. If the axis sits above the region, at y = k, the radius is k minus the function value, since the curve is below the line and the gap shrinks as the curve rises toward it. The volume is still π times the integral of radius squared dx, but radius now carries that shift.

This topic is the natural partner to basic disc and washer problems around the x-axis or y-axis: same slicing idea, same integral setup, just a relocated axis. It also sets up washers around shifted axes, where both an outer and inner radius need the same kind of adjustment. Getting comfortable with shifting the radius here makes every later volume problem, including those with two curves and an off-axis line, much more routine.