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

Section 6.2 Volumes 443
The cross-sectional area is
Asxd − s2 2 x 2 d 2 2 s2 2 xd 2
and so the volume of S is
V − y 1
Asxd dx
0
− y 1
fs2 2 x 2 d 2 2 s2 2 xd 2 g dx
0
− y 1
sx 4 2 5x 2 1 4xd dx
0
− F x 5
5 2 5 x 3
3 1 4 x 2
1
2G0
− 8
15
n
The solids in Examples 1–5 are all called solids of revolution because they are
obtained by revolving a region about a line. In general, we calculate the volume of a
solid of revo lution by using the basic defining formula
V − y b
Asxd dx or V − y d
Asyd dy
a
and we find the cross-sectional area Asxd or Asyd in one of the following ways:
• If the cross-section is a disk (as in Examples 1–3), we find the radius of the
disk (in terms of x or y) and use
A − sradiusd 2
• If the cross-section is a washer (as in Examples 4 and 5), we find the inner
radius r in and outer radius r out from a sketch (as in Figures 8, 9, and 10) and
compute the area of the washer by subtracting the area of the inner disk from
the area of the outer disk:
A − souter radiusd 2 2 sinner radiusd 2
c
r in
r out
FIGURE 10
The next example gives a further illustration of the procedure.
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