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

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

286 STEP 4. Review the Knowledge You Need to Score High

Example 1

Find the area of the regions bounded by the graphs off(x)=x^3 andg(x)=x.

(See Figure 13.3-9.)

– (^110)
(–1, 1)
(1, 1)
y
x
g(x)
f(x)
Figure 13.3-9
Step 1: Sketch the graphs off(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 the 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

.