5 Steps to a 5 AP Calculus BC 2019

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.