Introductory university-level calculus, linear algebra, abstract algebra, probability, statistics, and stochastic processes.
Riemann integral
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You may have learned that the integral is the inverse operation of differentiation. Here, we define the integral as the calculation of area. This approach has an advantage that can be easily extended to higher dimensions.
Definition (Partition of an interval)
The partition of the closed interval is a finite sequence of the form
Each is called a sub-interval. The mesh or norm of a partition is defined to be the maximum length of the sub-intervals:
Let be a bounded function on . Let us define the following quantities:
Remark. If is continuous on , has maximum and minimum values on each (the Extreme Value Theorem). Thus, and are the maximum and minimum values. If is not continuous, it may not have maximum or minimum values. Nevertheless, since we are assuming is bounded, supremum and infimum do exist. Thus and are well-defined. □
are called the upper Riemann sum and lower Riemann sum, respectively, with respect to the partition .
Remark. Upper or lower Riemann sums are also known as upper and lower Darboux sums. □
Each (or ) represents the area of a rectangle with the ``width'' of and the ``height'' of (or ). Note that (or ) can be negative. So this area is a signed area.
The Riemann sums approximate the signed area enclosed by the graph of , the -axis, and . Clearly,
Definition (Refinement of a partition)
Let and be partitions of . is said to be a refinement of if each is equal to some .
Remark. If and , then the sub-interval is partitioned into smaller sub-intervals
□
Example. Consider the interval .
is a partition of .
is another partition of and also a refinement of .
□
Lemma
Let be a bounded function on . Let be a partition of and be a refinement of . Then
where and are lower Riemann sums of with respect to and , respectively, and and are the corresponding upper Riemann sums.
Proof. Let be a sub-interval of , and be a sub-interval of such that .
Let and .
Then, because is at least as low as (since covers a wider range than ).
If and , then
Summing both sides over , we have
is similarly proved. ■
If we consider a sequence of finer partitions , we have
Therefore the sequence of the lower Riemann sums is monotone increasing with respect to refinements of partitions, and is bounded above by upper Riemann sums (e.g., , etc). By the continuity axiom of real numbers, exists ( means supremum over all possible partitions ). Similarly, exists. Clearly,
Based on the above observation, we define the definite integral as follows.
Definition (Definite integral)
Let be a bounded function on the closed interval . Let and be the lower and upper Riemann sums of with respect to the partition . The function is said to be Riemann-integrable, or simply integrable, if and we write
which we call the definite integral of on .
Example. Let us define the following function on (called the Dirichlet function):
This function is not Riemann-integrable. Take any sub-interval of any partition . Since rational numbers are dense, there is always at least one rational number in . Hence . Similarly, irrational numbers are dense so that . Therefore
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