Some volume problems are very difﬁcult to handle by the methods of Section 6.2. For
instance, let’s consider the problem of ﬁnding the volume of the solid obtained by rotating
about the axis the region bounded by
and
. (See Figure 1.) If we slice
perpendicular to the
y
axis, we get a washer. But to compute the inner radius and the outer
radius of the washer, we would have to solve the cubic equation
for
x
in
terms of
y
; that’s not easy.
Fortunately, there is a method, called the
method of cylindrical shells
, that is easier to
use in such a case. Figure 2 shows a cylindrical shell with inner radius
, outer radius
,
and height . Its volume
is calculated by subtracting the volume
of the inner cylinder
from the volume
of the outer cylinder:
If we let
(the thickness of the shell) and
(the average radius
of the shell), then this formula for the volume of a cylindrical shell becomes
and it can be remembered as
Now let
be the solid obtained by rotating about the axis the region bounded by
[where
],
and
, where
. (See Figure 3.)
We divide the interval
into
n
subintervals
of equal width
and let
be
the midpoint of the
i
th subinterval. If the rectangle with base
and height
is
rotated about the
y
axis, then the result is a cylindrical shell with average radius
, height
, and thickness
(see Figure 4), so by Formula 1 its volume is
Therefore, an approximation to the volume
of
is given by the sum of the volumes of
these shells:
V
±
²
n
i
±
1
V
i
±
²
n
i
±
1
2
±
x
i
f
³
x
i
´
²
x
S
V
V
i
±
³
2
x
i
´µ
f
³
x
i
´¶
²
x
²
x
f
³
x
i
´
x
i
f
³
x
i
´
µ
x
i
³
1
,
x
i
¶
x
i
²
x
µ
x
i
³
1
,
x
i
¶
µ
a
,
b
¶
FIGURE 3
ab
x
y
0
y=ƒ
x
y
0
y=ƒ
b
´
a
µ
0
x
±
b
y
±
0,
x
±
a
,
f
³
x
´
µ
0
y
±
f
³
x
´
y
S
V
±
[circumference][height][thickness]
V
±
2
rh
²
r
1
r
±
1
2
³
r
2
¶
r
1
´
²
r
±
r
2
³
r
1
±
2
r
2
¶
r
1
2
h
³
r
2
³
r
1
´
±
³
r
2
¶
r
1
´³
r
2
³
r
1
´
h
±
r
2
2
h
³
r
2
1
h
±
³
r
2
2
³
r
2
1
´
h
V
±
V
2
³
V
1
V
2
V
1
V
h
r
2
r
1
y
±
2
x
2
³
x
3
y
±
0
y
±
2
x
2
³
x
3
y
FIGURE 1
FIGURE 2
r
r¡
r™
Îr
h
y
x
0
2
1
y=2≈˛
x
L
=
?
x
R
=
?
1
Volumes by Cylindrical Shells
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View Full DocumentThis approximation appears to become better as
. But, from the deﬁnition of an inte
gral, we know that
Thus, the following appears plausible:
The volume of the solid in Figure 3, obtained by rotating about the
y
axis the
region under the curve
from
a
to
b
,is
The argument using cylindrical shells makes Formula 2 seem reasonable, but later we
will be able to prove it. (See Exercise 47.)
The best way to remember Formula 2 is to think of a typical shell, cut and ﬂattened as
in Figure 5, with radius
x
, circumference
, height
, and thickness
or
:
This type of reasoning will be helpful in other situations, such as when we rotate about
lines other than the
y
axis.
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 Spring '11
 jittat
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