-
method of cylindrical shells: solids obtained by revolving a region
about a line may be obtained by chopping into cylindrical shells, which are
then added up. Each cylindrical shell has volume given approximately by
or
where x is the radius of a shell. We add up these small volumes in the form of an integral, i.e.
where
.
Sometimes cylindrical shells is easier for computing volumes, sometimes disks or washers are easiest; it really depends on you and the particular problem. Do it any way which is easy for you! One reason for why one method may be easier than another is that the function given (for height, say, is expressed in terms of y, say, and difficult to express in terms of x): this may make one method easier than the other.
One thing you need to do to make sure that you're on the right track is to make a sketch of the situation, so that you're using the correct radius, limits of integration, and height!