Concept preview

What you will be able to do

1. 01Extend the washer method to revolution about a horizontal line $y=k$ or a vertical line $x=h$ that is not a coordinate axis, by measuring each radius as the distance from a curve to that line.
2. 02Correctly form the outer and inner radii as $R=\lvert\text{farther curve}-k\rvert$ and $r=\lvert\text{nearer curve}-k\rvert$, deciding which curve is farther from the axis of revolution before squaring.
3. 03Set up and evaluate $V=\pi\int\left(R^2-r^2\right)$ about a shifted line, keeping $\pi$ symbolic, and avoid the slip of measuring radii from the coordinate axis instead of from the given line.

Revolve a region about the horizontal line $y=k$. Each slice is still perpendicular to the axis, so it is vertical-to-horizontal in the same way as before, and each radius runs from the line $y=k$ straight to a boundary curve. The length of that radius is the vertical distance $\lvert\text{curve}-k\rvert$. If the whole region lies BELOW the line (every curve value is less than $k$), the distances are $k-\text{curve}$; if it lies ABOVE the line, the distances are $\text{curve}-k$. The outer radius $R$ belongs to whichever curve is FARTHER from the line, and the inner radius $r$ to the curve nearer the line.