Volume with Washer Method: Revolving Around Other Axes

The washer method finds volume by stacking thin rings and adding up their areas: π times (outer radius squared minus inner radius squared) times a thin thickness dx. When you revolve around the x-axis, the radius of each curve is just its y-value. Move the axis to some other horizontal line, like y = 5, and every radius has to be rewritten as the distance from that line to the curve: axis value minus the curve's y-value, or the curve's y-value minus the axis value, whichever is positive.

That distance is what decides outer versus inner, and it depends entirely on where the axis sits relative to the region. If the axis is above both curves, the curve that is lower in the picture is actually farther from the axis, so it becomes the outer radius, and the curve closer to the axis becomes the inner radius. If the axis is below both curves, the roles can flip again: now the curve that sits higher up is farther from the axis and becomes the outer radius. The only safe method is to pick a sample point, subtract the axis value from each curve's value, and see which difference is bigger.

This is the exact same shift that happens with the disc method, just with one radius instead of two. Shifting the axis there also means replacing y with (axis minus y) or (y minus axis) before squaring. Washers are just two discs subtracted, so getting the radius-shift right here is the same skill carried forward into shells and into any volume problem where the axis of revolution is not the one you started with.