Washer vs cylindrical shell method for computing volumes

[jaruta]

Hello,

Homework Statement

My problem regards the disk|washer, and cylindrical shell methods for finding volumes in single variable calc. My problem is basically am I understanding these two methods and their relationships properly.

Fundamentally, these methods are indentical, as we can arrive at the same solution whichever one we use. the difference is in the mechanics. At least that`s what I can tell from experience. The way in which I comprehend these methods is basically such that the washer method (which is essentially a corrolary to the disk method, no?) is applicable when the structure of the solid is such that one can infer the cross-sectional area; whereas, in the cyclidrical shell method one cannot. Moreover, because the rotation in the cylidrical shell method is parallel to the axis of revolution one would need to compute the max and min values and shift the solid to properly compute the volume once the area is revolved. Under this scenario, would it also be required to compute the thickness of the inner cylinder if it changed?

 

[Mark44]

Hello,

Homework Statement

My problem regards the disk|washer, and cylindrical shell methods for finding volumes in single variable calc. My problem is basically am I understanding these two methods and their relationships properly.

Fundamentally, these methods are indentical, as we can arrive at the same solution whichever one we use.

Merely arriving at the same numerical answer does not make these methods identical
. The geometric objects that are used are very different.

the difference is in the mechanics. At least that`s what I can tell from experience. The way in which I comprehend these methods is basically such that the washer method (which is essentially a corrolary to the disk method, no?)

Yes. A washer is a disk with a circular hole at its center.

is applicable when the structure of the solid is such that one can infer the cross-sectional area; whereas, in the cyclidrical shell method one cannot.

No, that's not the difference. In both methods you are finding the the volume of a typical volume element, and this volume comes from the area of some cross section.

Most (all?) problems can be done using either method, but often one method is more convenient to use.

Moreover, because the rotation in the cylidrical shell method is parallel to the axis of revolution one would need to compute the max and min values and shift the solid to properly compute the volume once the area is revolved. Under this scenario, would it also be required to compute the thickness of the inner cylinder if it changed?

I'm not clear on what you're asking here? Do you have an example in mind?