288 STEP 4. Review the Knowledge You Need to Score High
Step 1. Draw a sketch. (See Figure 12.4-14.)
y
x
x = y
x = y^2
1
0
(1,1)
Figure 12.4-14
Intersection points:y=x^2 ;x=y^2 ⇒y=±
√
x.
Setx^2 =
√
x⇒x^4 =x⇒x^4 −x= 0 ⇒x(x^3 −1)= 0 ⇒x=0orx= 1
x=0, y=0(0,0)
x=1, y=1(1,1).
Step 2. Determine the outer and inner radii of a washer, with outer radius:
x=√yand inner radius:x=y^2.
Step 3. Set up an integral.
V=π
∫ 1
0
(
(
√
y)^2 −
(
y^2
) 2 )
dy
Step 4. Evaluate the integral.
Enter
∫ (
π∗
(
(√y)∧ 2 −(y∧ 2 )∧ 2
)
, y,0,1
)
and obtain
3 π
10
.
The volume of the solid is
3 π
10