Chapter 5 | Integration
537
Finding the Total Area
Find the total area between f ( x ) = x −2 and the x -axis over the interval ⎡ ⎣ 0, 6 ⎤ ⎦ .
Solution Calculate the x -intercept as (2, 0) (set y =0, solve for x ). To find the total area, take the area below the x -axis over the subinterval [0, 2] and add it to the area above the x -axis on the subinterval ⎡ ⎣ 2, 6 ⎤ ⎦ ( Figure 5.22 ).
Figure 5.22 The total area between the line and the x-axis over ⎡ ⎣ 0, 6 ⎤ ⎦ is A 2 plus A 1 .
We have
6 | ( x −2) | dx = A 2 + A 1 .
∫
0
Then, using the formula for the area of a triangle, we obtain A 2 = 1 2 bh = 1
2 ·2·2=2
A 1 = 1 2
bh = 1
2 ·4·4=8.
The total area, then, is
A 1 + A 2 =8+2=10.
Find the total area between the function f ( x ) =2 x and the x -axis over the interval [−3, 3].
5.10
Properties of the Definite Integral The properties of indefinite integrals apply to definite integrals as well. Definite integrals also have properties that relate to the limits of integration. These properties, along with the rules of integration that we examine later in this chapter, help us manipulate expressions to evaluate definite integrals.
Rule: Properties of the Definite Integral 1.
a
(5.9)
∫
f ( x ) dx =0
a
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