5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 273

Example 1

Find the area of the regions bounded by the graphs of f(x)=x^3 andg(x)=x. (See
Figure 12.3-9.)

–1 0 1

(–1,1)

(1,1)

y

x

g(x)

f(x)

Figure 12.3-9

Step 1. Sketch the graphs of f(x) andg(x).

Step 2. Find the points of intersection.

Setf(x)=g(x)

x^3 =x

⇒x(x^2 −1)= 0

⇒x(x−1)(x+1)= 0

⇒x=0, 1, and− 1.

Step 3. Set up integrals.

Area=

∫ 0

− 1

(f(x)−g(x))dx+

∫ 1

0

(g(x)−f(x))dx

=

∫ 0

− 1

(

x^3 −x

)

dx+

∫ 1

0

(

x−x^3

)

dx

=

[

x^4

4

−

x^2

2

] 0

− 1

+

[

x^2

2

−

x^4

4

] 1

0

= 0 −

(

(− 1 )^4

4

−

(− 1 )^2

2

)

+

(

12

2

−

14

4

)

− 0

=−

(

−

1

4

)

+

1

4

=

1

2

.