Areas, Volumes, and Arc Lengths 287
Example 2
Using the Washer Method and a calculator, find the volume of the solid generated by
revolving the region in Example 1 about the liney=2.
Step 1. Draw a sketch. (See Figure 12.4-13.)
y = x^3
y = x
y = 2
x
0
y
Figure 12.4-13
Step 2. Determine the outer and inner radii of a washer.
The outer radius=(2−x^3 ) and inner radius=(2−x).
Step 3. Set up an integral.
V=π
∫ 1
0
[(
2 −x^3
) 2
−( 2 −x)^2
]
dx
Step 4. Evaluate the integral.
Enter
∫ (
π∗
(
( 2 −x∧ 3 )∧ 2 −(2−x)∧ 2
)
, x,0,1
)
and obtain
17 π
21
.
The volume of the solid is
17 π
21
.
Example 3
Using the Washer Method and a calculator, find the volume of the solid generated by
revolving the region bounded byy=x^2 andx=y^2 about they-axis.