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

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

Areas and Volumes 283

Step 3: Evaluate the integrals.

∫− 1

− 2

(

x^2 − 1

)

dx=

x^3

3

−x

]− 1

− 2

=

2

3

−

(

−

2

3

)

=

4

3

∣

∣∣

∣

∫ 1

− 1

(

x^2 − 1

)

dx

∣

∣∣

∣=

∣∣

∣∣

∣

x^3

3

−x

] 1

− 1

∣∣

∣∣

∣

=

∣

∣∣

∣−

2

3

−

(

2

3

)∣∣

∣∣=

∣

∣∣

∣−

4

3

∣

∣∣

∣=

4

3

∫ 2

1

(

x^2 − 1

)

dx=

x^3

3

−x

] 2

1

=

2

3

−

(

−

2

3

)

=

4

3

Thus, the total area=

4

3

+

4

3

+

4

3

=4.

Note that since f(x)=x^2 −1 is an even function, you can use the symmetry of the

graph and set area = 2

(∣∣

∣∣

∫ 1

0

f(x)dx

∣∣

∣∣+

∫ 2

1

f(x)dx

)

.

An alternate solution is to find the area using a calculator:

Enter

∫

(abs(x∧ 2 − 1 ),x,−2, 2)and obtain 4.

Example 3

Find the area of the region bounded byx=y^2 ,y=−1, andy=3. (See Figure 13.3-5.)

3

0

• 1 x = y^2

x

y

Figure 13.3-5

Area =

∫ 3

− 1

y^2 dy=

y^3

3

] 3

− 1

=

33

3

−

(−1)^3

3

=

28

3

.