5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:1

Areas and Volumes 299

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 13.4-12.)

y

x

0

y =

x

y =

x^3

(1, 1)

Figure 13.4-12

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.
The outer radius=xand 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.