How to Find Volume by Spinning: Washer Method
TL;DR: Spin a region that does not touch the axis and you get a solid with a hole down the middle — like a stack of washers. The washer method handles exactly this case. Each slice is an outer disk minus an inner disk, so you integrate pi times outer radius squared minus inner radius […]
How to Find Volume by Spinning: Shell Method
TL;DR: Sometimes slicing perpendicular to the axis is a mess — the radii change weirdly or the bounds get ugly. The shell method takes a different approach by integrating thin cylindrical shells parallel to the axis instead. Each shell contributes 2 pi times its radius times its height times dx. Use it when slicing along […]