Shell method VS washer/disk method

[Pixleateit]

i'm confused right now about this, when would it be appropriate to use the shell method rather than using the washer or disk method when it comes to looking for the volume of a rotating object?

 

[HallsofIvy]

Often either can be used. It really depends upon whether it is easier to write y as a function of x or x as a function of y and whether the axis of rotation is parallel to the x or y axis.

If you have y as a function of x, and the axis of rotation is parallel to the x-axis, then your "disk radius" will depend on y but you will be integrating with respect to x so you will, as you want, be integrating a function of x with respect to x. That would be using the disk method.

If you have y as a function of x, and the axis of rotation is parallel to the y-axis, trying to use the disk method, your "disk radius" will depend on x but you want to integrate with respect to y and x is NOT a function of y. That will cause difficulties. Use the shell method here.

If you have x as a function of y, reverse the two above. Of course, if you can easily write both y as a function of x and x as a function of y, it doesn't matter.

This makes much more sense for a specific problem than as a general rule!

 

[mathwonk]

washers/discs are more suitable in terms of x,y,z coords, roughly, and shells in terms of polar coords, so yes i would say to use the latter on a problem with rotational symmetry.

i.e. they are both the same method just expressed in different coords. so use whichever one suits the problem, as halls suggests.