The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Volumes of revolution: disks and washers
Choose the axis, radius, variable and matching limits.
Use disks or washers as appropriate, without double-counting overlapping rotations.
Solve inverse volume questions and simple limiting-volume problems.
About the x-axis use \(V=\pi\int y^2\,dx\); about the y-axis use \(V=\pi\int x^2\,dy\). A washer uses \(\pi(R^2-r^2)\), not \(\pi(R-r)^2\). Radii are distances to the axis; check whether the region crosses it.