8.12 Volume with Washer Method: Revolving Around Other Axes

Setting up the washer for shifted axes

When revolving around a horizontal line y = k (not the x-axis), the radii shift by k: outer radius = R(x) − k, inner radius = r(x) − k. Volume = π∫[a,b] (R(x)−k)² − (r(x)−k)² dx. For vertical lines x = h, integrate with respect to y using (R(y)−h) and (r(y)−h).

Identifying outer vs inner radius

Sketch the region and axis of rotation first. The outer radius is the distance from the axis to the curve farther away; the inner radius is the distance to the closer curve. Always subtract the axis location from each function's value — order matters for setting up correctly, but since it's squared, sign errors often cancel out in area but not in radius identification (which curve is farther).

Choosing the integration variable

Rotating around a horizontal axis (y = k) usually means integrating with respect to x; rotating around a vertical axis (x = h) usually means integrating with respect to y. If the given functions aren't naturally in terms of the needed variable, solve for x in terms of y (or vice versa) before setting up radii.

Key terms

Washer volume around y = k

V = π∫[a,b] (R(x)−k)² − (r(x)−k)² dx, where R and r are outer/inner function values.

Washer volume around x = h

V = π∫[c,d] (R(y)−h)² − (r(y)−h)² dy, integrating with respect to y.

Outer radius

Distance from the axis of rotation to the curve farther from that axis.

Inner radius

Distance from the axis of rotation to the curve closer to that axis.

Choosing integration variable

Horizontal axis (y = k) → integrate dx; vertical axis (x = h) → integrate dy.

Try an AP-style question

The region between y = x and y = x² is revolved about y = −1. What is the volume?

1. 7π/15
2. 2π/15
3. π/3
4. 8π/15