By Rick Stoll |
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The following properties are easy to check: Theorem. If f (x) and g(x) are defined and continuous on [a, b], except maybe at a finite number of points, then we have the following linearity principle for the integral:
The next results are very useful in many problems. Theorem. If f (x) is defined and continuous on [a, b], except maybe at a finite number of points, then we have
The property (ii) can be easily illustrated by the following picture:
Remark. It is easy to see from the definition of lower and upper sums that if f (x) is positive then f (x) dx 0. This implies the following
If
f (x)
g(x) for
x
[a, b]
f
(x) dx
g(x) dx
.
Example. We have
(x2
- 2x)dx = x2 dx
- 2x dx
.
We have seen previously that
x2 dx
=
and x dx
=
.
Hence
(x2
- 2x)dx =
- 2
= -
.
Exercise 1. Given that
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