James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 6.3 Volumes by Cylindrical Shells 453
SOLUtion Figure 10 shows the region and a cylindrical shell formed by rotation about
the line x − 2. It has radius 2 2 x, circumference 2s2 2 xd, and height x 2 x 2 .
y
y
y=x-≈
x=2
FIGURE 10
0
x
0
x
1 2 3 4
2-x
x
The volume of the given solid is
V − y 1
2s2 2 xdsx 2 x 2 d dx
0
− 2 y 1
sx 3 2 3x 2 1 2xd dx
0
− 2F x 4
4 2 x 3 1 x 2G0
− 2
1
n
Disks and Washers versus Cylindrical Shells
When computing the volume of a solid of revolution, how do we know whether to use
disks (or washers) or cylindrical shells? There are several considerations to take into
account: Is the region more easily described by top and bottom boundary curves of the
form y − f sxd, or by left and right boundaries x − tsyd? Which choice is easier to work
with? Are the limits of integration easier to find for one variable versus the other? Does
the region require two separate integrals when using x as the variable but only one integral
in y? Are we able to evaluate the integral we set up with our choice of variable?
If we decide that one variable is easier to work with than the other, then this dictates
which method to use. Draw a sample rectangle in the region, corresponding to a crosssection
of the solid. The thickness of the rectangle, either Dx or Dy, corresponds to the
integration variable. If you imagine the rectangle revolving, it becomes either a disk
(washer) or a shell.
1. Let S be the solid obtained by rotating the region shown in the
figure about the y-axis. Explain why it is awkward to use slicing
to find the volume V of S. Sketch a typical approximating shell.
What are its circumference and height? Use shells to find V.
y
y=x(x-1)@
2. Let S be the solid obtained by rotating the region shown in the
figure about the y-axis. Sketch a typical cylindrical shell and
find its circumference and height. Use shells to find the volume
of S. Do you think this method is preferable to slicing? Explain.
y
y=sin{≈}
0 1 x
0
œ„π x
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