286 STEP 4. Review the Knowledge You Need to Score High
About a linex=h:
V=π
∫b
a
[
(f(x)−h)^2 −(g(x)−h)^2
]
dx.
About a liney=k:
V=π
∫d
c
[
(p(y)−k)^2 −(q(y)−k)^2
]
dy.
Example 1
Using the Washer Method, find the volume of the solid generated by revolving the region
bounded byy=x^3 andy=xin the first quadrant about thex-axis.
Step 1. Draw a sketch. (See Figure 12.4-12.)
y
0 x
y =
x
y =
x^3
(1,1)
Figure 12.4-12
To find the points of intersection, setx=x^3 ⇒x^3 −x=0orx(x^2 −1)=0, or
x=−1, 0, 1. In the first quadrantx=0, 1.
Step 2. Determine the outer and inner radii of a washer whose outer radius=x, and inner
radius=x^3.
Step 3. Set up an integral.
V=
∫ 1
0
[
x^2 −
(
x^3
) 2 ]
dx
Step 4. Evaluate the integral.
V=
∫ 1
0
(
x^2 −x^6
)
dx=π
[
x^3
3
−
x^7
7
] 1
0
=π
(
1
3
−
1
7
)
=
4 π
21
Verify your result with a calculator.