288 STEP 4. Review the Knowledge You Need to Score High
Step 1. Draw a sketch. (See Figure 12.4-14.)yxx = yx = y^210(1,1)Figure 12.4-14Intersection 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∫ 10(
(√
y)^2 −(
y^2) 2 )
dyStep 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