Shell Vs Washer Method?

Question

Can someone please explain how to know when to use the shell vs the washer method. the way I understand it one is taking the volume of circles perpendicular axis and one of cylinders parallel axis. If this reasoning is right so far, what is the difference between taking the volume of the perpendicular or the parallel? Why are the shapes different? And how would you know in a word problem or something when to use which?

Thank you!!

Answer 1

Case 1:

desmos.com/calculator/m8le2frgik

Case 2:

desmos.com/calculator/hp2xvrprv6

Case 3:

desmos.com/calculator/omn6rlzivj

Answer 2

Just physically, there are two directions you can integrate. Along the axis, and away from the axis. (The third direction is around the axis, but that is harder to do without higher level stuff.)

1. If you go along the axis, the cross-sections are planes. When you intersect the plane with the round ("revolved") shape, you get a disk.
2. If you progress out of the axis, the cross-sections are inherently cylinders. That's it.

Both ways always work for every problem.

Just use basic geometry and set up slices like any other volume problem. There's nothing special about the roundness, and the "methods" are just formulas so you don't have to know what you're doing. But that's not helpful since you have to set up the geometry to know what to plug in the formula anyway. I would forget them entirely, then slap on the label they want afterward.

If you want more info about which one you should pick, it actually has nothing to do with the word problem or even the revolution. Just look at the 2D region and decide like you would any other. Do you want to integrate x or y? The decision is based on the shape (will corners force a piecewise function?) and the functions (e.g. is it hard to solve for x?) But don't overthink it. Both ways always work. It's up to taste. Doing something is better than nothing.