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

450 Chapter 6 Applications of Integration
r 2 , and height h. Its volume V is calculated by subtracting the volume V 1 of the inner
cylinder from the volume V 2 of the outer cylinder:
V − V 2 2 V 1
− r 2 2 h 2 r 2 1 h − sr 2 2
2 r 2 1 dh
− sr 2 1 r 1 dsr 2 2 r 1 dh
− 2 r 2 1 r 1
2
hsr 2 2 r 1 d
If we let Dr − r 2 2 r 1 (the thickness of the shell) and r − 1 2 sr 2 1 r 1 d (the average radius
of the shell), then this formula for the volume of a cylindrical shell becomes
1
V − 2rh Dr
and it can be remembered as
V − [circumference][height][thickness]
Now let S be the solid obtained by rotating about the y-axis the region bounded by
y − f sxd [where f sxd > 0], y − 0, x − a, and x − b, where b . a > 0. (See Figure 3.)
y
y=ƒ
y
y=ƒ
0
a b x
0
a
b
x
FIGURE 3
We divide the interval fa, bg into n subintervals fx i21 , x i g of equal width Dx and let x i
be the midpoint of the ith subinterval. If the rectangle with base fx i21 , x i g and height f sx i d
is rotated about the y-axis, then the result is a cylindrical shell with average radius x i ,
height f sx i d, and thickness Dx (see Figure 4). So by Formula 1 its volume is
V i − s2x i df f sx i dg Dx
y
y=ƒ
y
y=ƒ
y
y=ƒ
0 a b x
0
a
b
x
0
a
b
x
FIGURE 4
x i-1
x–
i
x i
Therefore an approximation to the volume V of S is given by the sum of the volumes of
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