Page 242 - Calculus Demystified

Applications of the Integral
CHAPTER 8

Fig. 8.18 229

y

h

r x
Fig. 8.19
EXAMPLE 8.9
Use the method of cylindrical shells to calculate the volume of the solid
2
enclosed when the curve y = x ,1 ≤ x ≤ 3, isrotated about the y-axis.
SOLUTION
2
As usual, we think of the region under y = x and above the x-axis as
composed of vertical segments or strips. The segment at position x has height
2
2
x . Thus, in this instance, h = x , r = x, and the volume of the cylinder is
2
2πx · x · x. As a result, the requested volume is
3

2
V = 2πx · x dx.
1
We easily calculate this to equal

3 4 3 3 4 1 4
3
V = 2π · x dx = 2π x = 2π − = 40π.
1 4 1 4 4