Volume with Disc Method: Revolving Around Other Axes

The disc method always builds volume out of thin circular slices stacked along the axis of rotation. Each slice is a disc with some radius, area π r², and thickness dx. None of that changes when you move the axis away from the x-axis. What changes is only what counts as the radius: it's no longer just the height of the curve, f(x), but the distance from the curve down (or up) to wherever the new axis sits.

To write that radius, find the gap between the curve and the line. If you're revolving around y = -k, and the curve sits above that line, the radius is f(x) + k, because you have to travel up from y = -k to y = 0 (a distance k) and then up to f(x) (a distance f(x)). If the axis were above the region instead, you would subtract the curve from the axis height. Once you have the radius as a function of x, square it, multiply by π, and integrate over the interval just like always: V = integral of π [radius(x)]² dx.

This is the same disc method from revolving around the x-axis, just with one extra algebra step before you integrate. It sets up the next idea directly: when the region between two curves revolves around a shifted axis, you subtract two radii instead of building one, which turns the disc method into the washer method. Getting the radius distance right here is what makes that next step manageable.