Department of Mathematics, Allegheny College, Meadville, PA 16335.
The American Mathematical Monthly, February 1984, Volume 91, Number 2, p. 139.
In most calculus books there is little effort given to showing that the cylindrical shell.
method and disk method give the same value when computing the volume of a solid.
of revolution. Indeed it is not obvious that these two distinct methods should give.
the same result. In some texts this is demonstrated when the trapezoid bounded by the.
x-axis, and is revolved about the y-axis.
In this paper we shall show that the cylindrical shell and disk methods give the same.
value if the region revolved about the y-axis is bounded by and.
the x-axis, provided is a differentiable function on and is one-to-one.
The proof is simple and uses two theorems which the students have recently learned.
(substitution formula and integration by parts.) This proof can easily be included in a.
calculus course.
Consider the solid of revolution K produced by revolving the region bounded by.
and the x-axis, about the y-axis. We use the shell method, which.
involves summing the volumes of cylindrical shells, to define the volume of K to be.
If is differentiable on and hence continuous.
there, this limit exists and is equal to.
Suppose the region is bounded by the function and the y-axis. In.
the disk method, which involves summing the volumes of disks, we consider.
If is continuous on this limit exists and is equal to.
THEOREM. Let and let be differentiable, nonnegative and.
on with and where * Also let be continuous on.
and let iff If R is the region bounded by the x-axis,.
and and R is revolved about the y-axis, then the value obtained by using.
the disk method is equal to the value obtained by using the cylindrical shell method.
Equivalently.
Proof: The region R can also be described as the region bounded by.
and where.
We observe from the way that is defined that it is continuous on If we.
evaluate the volume obtained by revolving the region R about the y-axis by using the.