Page 243 - Calculus Demystified

CHAPTER 8
230
EXAMPLE 8.10 Applications of the Integral
Use the method of cylindrical shells to calculate the volume enclosed when
2
the curve y = x ,0 ≤ x ≤ 3, isrotated about the x-axis(Fig. 8.20).

y

x

Fig. 8.20

SOLUTION
We reverse, in our analysis, the roles of the x- and y-axes. Of course y
ranges from 0 to 9. For each position y in that range, there is a segment stretch-
√ √
ing from x = y to x = 3. Thus it has length 3 − y. Then the cylinder
generated when this segment (thickened to a strip of width y) is rotated
about the x-axis has volume

√
V(y) = 2πy · 3 − y y.
The aggregate volume is then

9

√
V = 2πy · 3 − y dy
0
9

= 2π · 3y − y 3/2 dy
0
2 5/2 9
3y y
= 2π · − dy
2 5/2 0