Chapter 5 | Integration
539
1 ⎛
−2 1
−2 1
−2 1 −2 1
⎝ −3 x 3 +2 x +2 ⎞
∫
⎠ dx = ∫
−3 x 3 dx + ∫
2 xdx + ∫
2 dx
−2
−2 1
−2 1
=−3 ∫
x 3 dx +2 ∫
xdx + ∫
2 dx .
5.11 Use the properties of the definite integral to express the definite integral of f ( x ) =6 x 3 −4 x 2 +2 x −3 over the interval [1, 3] as the sum of four definite integrals.
Example 5.12 Using the Properties of the Definite Integral
If it is known that ∫ 0 8
f ( x ) dx =10 and ∫ 0 5
f ( x ) dx =5, find the value of ∫ 5 8
f ( x ) dx .
Solution By property 6.,
b
c
f ( x ) dx + ∫ c b
∫ a
f ( x ) dx = ∫
f ( x ) dx .
a
Thus,
8
f ( x ) dx = ∫ 0 5
8
∫
f ( x ) dx + ∫
f ( x ) dx
0
5
10 = 5+ ∫ 5 8
f ( x ) dx
8
5 = ∫
f ( x ) dx .
5
5.12
If it is known that ∫ 1 5
f ( x ) dx =−3 and ∫ 2 5
f ( x ) dx =4, find the value of ∫ 1 2
f ( x ) dx .
Comparison Properties of Integrals A picture can sometimes tell us more about a function than the results of computations. Comparing functions by their graphs as well as by their algebraic expressions can often give new insight into the process of integration. Intuitively, we might say that if a function f ( x ) is above another function g ( x ), then the area between f ( x ) and the x -axis is greater than the area between g ( x ) and the x -axis. This is true depending on the interval over which the comparison is made. The properties of definite integrals are valid whether a < b , a = b , or a > b . The following properties, however, concern only the case a ≤ b , and are used when we want to compare the sizes of integrals.
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