5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 269

Step 3. Evaluate the integrals.

∣∣

∣∣

∫ 1

0

(x−1)^3 dx

∣∣

∣∣=

∣∣

∣

∣∣

(x−1)^4

4

] 1

0

∣∣

∣

∣∣=

∣∣

∣∣−^1

4

∣∣

∣∣=^1

4

∫ 2

1

(x−1)^3 dx=

(x−1)^4

4

] 2

1

=

1

4

Thus, the total area is

1

4

+

1

4

=

1

2

.

Another solution is to find the area using a calculator.

Enter

∫ (

abs

(

(x− 1 )∧ 3

)

,x,0,2

)

and obtain

1

2

.

Example 2

Find the area of the region bounded by the graph of f(x)=x^2 −1, the linesx=−2 and
x=2, and thex-axis.

Step 1. Sketch the graph of f(x). See Figure 12.3-4.

(+) (+)

–2 –1 (^02) (–) 1
y
x
f(x)
Figure 12.3-4
Step 2. Set up integrals.
Area=
∫− 1
− 2
f(x)dx+
∣∣
∣∣
∫ 1
− 1
f(x)dx
∣∣
∣∣+
∫ 2
1
f(x)dx.