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

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

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

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 off(x). (See Figure 13.3-4.)

(+) (+)

• 2 – 1 0 1 2
  (–)

y

x

f(x)

Figure 13.3-4

Step 2: Set up the integrals.

Area=

∫− 1

− 2

f(x)dx+

∣∣

∣∣

∫ 1

− 1

f(x)dx

∣∣

∣∣+

∫ 2

1

f(x)dx.