270 STEP 4. Review the Knowledge You Need to Score High
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: 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 12.3-5.
3
0
–1 x = y^2
x
y
Figure 12.3-5
Area =
∫ 3
− 1
y^2 dy=
y^3
3
] 3
− 1