274 STEP 4. Review the Knowledge You Need to Score High
Note: You can use the symmetry of the graphs and let area = 2
∫ 1
0
(
x−x^3
)
dx.
An alternate solution is to find the area using a calculator. Enter
∫
(abs(x∧ 3 −x),x,−1, 1)
and obtain
1
2
.
Example 2
Find the area of the region bounded by the curvey=ex, they-axis and the liney=e^2.
Step 1. Sketch a graph. (See Figure 12.3-10.)
y = ex
y = e^2
x
y
1
0 12
Figure 12.3-10
Step 2. Find the point of intersection. Sete^2 =ex⇒x=2.
Step 3. Set up an integral:
Area =
∫ 2
0
(e^2 −ex)dx=(e^2 )x−ex]^20
=(2e^2 −e^2 )−(0−e^0 )
=e^2 + 1.
Or using a calculator, enter
∫
((e∧ 2 −e∧x),x,0,2)and obtain (e^2 +1).