Page 238 - Calculus Demystified

Applications of the Integral
CHAPTER 8

Fig. 8.14 225

There are two main methods for calculating volumes of solids of revolution:
the method of washers and the method of cylinders. The first of these is really an
instance of volume by slicing, just as we saw in the last section. The second uses a
different geometry; instead of slices one uses cylindrical shells. We shall develop
both technologies by way of some examples.
8.2.1 THE METHOD OF WASHERS

EXAMPLE 8.5
A solid is formed by rotating the triangle with vertices (0, 0), (2, 0), and (1, 1)
about the x-axis. See Fig. 8.15. What is the resulting volume?

(1,1)

(0,0) (2,0)

Fig. 8.15

SOLUTION
For 0 ≤ x ≤ 1, the upper edge of the triangle has equation y = x. Thus
the segment being rotated extends from (x, 0) to (x, x). Under rotation, it will
2
generate a disk of radius x, and hence area A(x) = πx . Thus the volume
generated over the segment 0 ≤ x ≤ 1is
1

2
V 1 = πx dx.
0
Similarly, for 1 ≤ x ≤ 2, the upper edge of the triangle has equation y =
2 − x. Thus the segment being rotated extends from (x, 0) to (x, 2 − x). Under