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
Example. We have
Exercise 1. Given that
Exercise 2. Let f (x) be defined and continuous on [a, b].
Assume that f (x) is positive. Show that the function
Exercise 3. Let f (x) and g(x) be two functions defined
and continuous on [a, b]. Show that
For more on the Area Problem, click HERE.
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