8.10 Volume with Disc Method: Revolving Around Other Axes

The disc method also works when the axis of rotation is not the x-axis or y-axis. The idea is exactly the same, but now the radius must be measured as the distance from the curve to the new axis. If the axis is horizontal, such as y=3, then the radius is a vertical distance. If the axis is vertical, such as x=-2, then the radius is a horizontal distance. The volume formula stays ${V=\int \pi r^2}$, but the expression for r changes.

You should always think “distance to the axis” rather than trying to memorize separate formulas for every possible line of rotation. If the curve is below a horizontal axis, the radius may be “axis minus function.” If it is above, the radius may be “function minus axis.” The same idea works for vertical axes. This lesson mainly tests whether you can translate geometric distance into the correct radius expression before squaring it and integrating.