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Objective: This demo builds a toolbox of teaching aids to illustrate various aspects of volume calculations using the washer method. Several props are used to demonstrate the geometric ideas of "washers" to obtain an approximation of the volumes of solids of revolution. A collection of animations is included which can be run on a number of platforms. Level: This demo can be presented in any course in which calculation of volumes of solids of revolution using the washer method is introduced. Prerequisites: Students should be familiar with areas of planar regions using approximations by Riemann sums and limits that lead to the definite integral. Prior knowledge of basic ideas concerning volumes is useful. Students should also be familiar with the disk method for computing volumes of solids of revolution. Platform: Mathematica notebooks that illustrate volumes modeled using the washer method are given. A gallery of animations that run in web browsers is provided. Several physical props are suggested that are useful for helping students understand the geometric concepts involving the washer method. Instructor's Notes: For this demo we focus on regions revolved about the x-axis or y-axis such that the resulting solid of revolution has a "hole," as illustrated in Figure 1.
If we slice the solid perpendicular to the axis of revolution as in Figure 2, the resulting solid resembles a washer, hence the technique for calculating the volume of the solid of revolution is called the washer method.
Figure 2. The washer method is based upon approximating the volume of the solid by adding the volume of the individual washers. The approximation process involves generating a partition and constructing the washers; the animation in Figure 3 shows the generation of a partition and construction of one of the approximating washers.
Figure 3. Some Useful Props To motivate the ideas central to the washer method, there are several props in addition to real washers that can be useful as visualization tools. LIFESAVERS candies (Figure 4) are easy to bring to class and share with students as the discussion begins. Before indulging, it is easy for students to hold the candy and think about how one could compute the volume of the candy by imagining that the candy is a disk and then subtracting the "hole."
Figure 4. Some tasteful props include angel food or bundt cake (any cake baked in a tube pan). By slicing the cake perpendicular to the hole, nice examples of washers (Figure 5) can be eaten later.
Figure 5. Another use of the angel cake model is to illustrate differences in using washers to approximate volumes and using shells. Figure 6 displays a cake washer and a cake shell. Note that the washer is sliced perpendicular to the hole (axis of revolution) while the orientation of the shell is parallel to the hole (axis of revolution).
Figure 6. CD-R disks often are sold in spindles of 50 or 100. The CDs on a spindle, shown in Figure 7, provide an easy to use way to illustrate how the washers stack up to form a larger solid.
Figure 7. Approximating the Volume As in the disk method, the washer method approximates the volume of the solid by adding the volumes of typical slices. We begin by determining the volume of a typical slice. Figure 8 illustrates how a washer can be generated from a disk. We begin with a disk with radius rout and thickness h. A smaller concentric disk with radius rin is removed from the original disk. The resulting solid is a washer.
Figure 8. The volume of the washer is calculated by subtracting the volume of the inner disk from the volume of the larger disk. Thus,
For a solid of revolution generated by revolving a region about the x-axis, the inner and outer radii are functions of x. Consider a region for which the upper boundary is the graph of y = f(x) and lower boundary is the graph of y = g(x) for x = a to x = b (Figure 9).
Figure 9. The
region is partitioned into n parts with width
The ith washer and its volume are shown in Figure 11.
Figure 11. By allowing the thickness of each washer to become very small and summing up the volumes, we obtain the definite integral representation for the washer method:
For a solid of revolution generated by revolving a region about the y-axis, the inner and outer radii are functions of y. Consider a region for which the right boundary is the graph of x = F(y) and left boundary is the graph of x = G(y) for y = c to y = d (Figure 12).
Figure 12. The
region is partitioned into n parts with thickness
Figure 13. The ith washer and its volume are shown in Figure 14.
Figure 14. By allowing the thickness of each washer to become very small and summing up the volumes, we obtain the definite integral representation for the washer method:
When we developed the disk method for computing volumes we illustrated the generation of the disks by revolving a rectangle about an axis. In a similar way, a washer is generated by revolving a rectangle about the axis of revolution. This is shown in the animations in Figure 15.
Figure 15. The animation in Figure 16 illustrates the sequence of steps involved with the washer method for computing the volume of the solid of revolution generated by revolving a region in the first quadrant bounded by the graphs of y = 1, x = 0, and y = x2 about the x-axis. First, the region is partitioned and a typical washer is drawn. Approximating half-washers are drawn. To complete the visualization, the approximating washers are produced. Finally, the solid of revolution is generated.
Figure 16. The animation in Figure 17
illustrates the sequence of steps involved with the washer method for
computing the volume of the solid of revolution generated by revolving a
region in the first quadrant bounded by the graphs of y = x2
and
Figure 17. A gallery of sample demos for illustrating the washer method for volumes of solids of revolution is available by clicking on WASHER-METHOD-GALLERY. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps of the washer method. The demos provide a variety of animations for some common examples. Also included is a step-by-step narrative script of the displays in the animations. Classroom Activities:
Technology Resources: There are a variety of resources that employ calculators or software for illustrating and computing volumes of solids of revolution. Following is a sample of such resources which can be located using a search engine. We have chosen ones that relate to the washer method.
Credits: This demo
was developed by and is included in the Demos with Positive Impact collection with her permission.
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LFR 5/15/04 Last updated 5/24/2006 DRH
.
,
where i = 1, 2, ..., n. Let xi be a value in the ith
interval [xi-1,xi]. The outer radius is
given by rout = f(xi) and the inner radius is rin
= g(xi)
(Figure 10).
.
,
where i = 1, 2, ..., n. Let yi be a value in the ith
interval [yi-1,yi]. The outer radius is
given by rout = F(yi) and the inner radius is rin
= G(yi)
(Figure 13).
.
about the y-axis First, the region is partitioned
and a typical washer is drawn. Approximating half-washers are
drawn. To complete the visualization, the approximating washers are
produced. Finally, the solid of revolution is generated.
