Riemann integral

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 [a,b] is a finite sequence Δ={xn} of the form

a=x0<x1<x2<<xn1<xn=b.

Each [xi,xi+1] is called a sub-interval. The mesh or norm of a partition is defined to be the maximum length of the sub-intervals:

max{(xi+1xi)i=0,1,,n1}.


Let f(x) be a bounded function on [a,b]. Let us define the following quantities:

Mi=sup{f(x)xixxi+1},mi=inf{f(x)xixxi+1}.

Remark. If f(x) is continuous on [a,b], f(x) has maximum and minimum values on each [xi,xi+1] (the Extreme Value Theorem). Thus, Mi and mi are the maximum and minimum values. If f(x) is not continuous, it may not have maximum or minimum values. Nevertheless, since we are assuming f(x) is bounded, supremum and infimum do exist. Thus Mi and mi are well-defined. □

See alsoContinuity of a function for the extreme value theorem.

Now, 

SΔ=i=0n1Mi(xi+1xi),sΔ=i=0n1mi(xi+1xi)

are called the upper Riemann sum and lower Riemann sum, respectively, with respect to the partition Δ.

Remark. Upper or lower Riemann sums are also known as upper and lower Darboux sums. □

Each mi(xi+1xi) (or Mi(xi+1xi)) represents the area of a rectangle with the ``width'' of xi+1xi and the ``height'' of mi (or Mi). Note that mi (or Mi) can be negative. So this area is a signed area.

The Riemann sums approximate the signed area enclosed by the graph of y=f(x), the x-axis, x=a and x=b. Clearly,

sΔSΔ.

Definition (Refinement of a partition)

Let Δ={x0,x1,,xn} and Δ={x0,x1,,xm} be partitions of [a,b]. Δ is said to be a refinement of Δ if each aiΔ is equal to some ajΔ.

Remark. If xi=xj and xi+1=xj+k, then the sub-interval [xi,xi+1] is partitioned into smaller sub-intervals

[xj,xj+1],[xj+1,xj+2],,[xj+k1,xj+k].

Example. Consider the interval I=[0,1].

  • Δ={0,0.5,1} is a partition of I
  • Δ={0,0.2,0.5,0.7,0.9,1} is another partition of I and also a refinement of Δ.

Lemma

Let f(x) be a bounded function on [a,b]. Let Δ be a partition of [a,b] and Δ be a refinement of Δ. Then
sΔsΔ,SΔSΔ
where sΔ and sΔ are lower Riemann sums of f(x) with respect to Δ and Δ, respectively, and SΔ and SΔ are the corresponding upper Riemann sums.
Proof. Let Ii=[xi,xi+1] be a sub-interval of Δ, and Ij=[xj,xj+1] be a sub-interval of Δ such that IjIi
Let mi=inf{f(x)xIi} and mj=inf{f(x)xIj}.
Then, mimj because mi is at least as low as mj (since Ii covers a wider range than Ij).
If xi=xj and xi+1=xj+k, then
mi(xi+1xi)=mil=0k1(xj+l+1xj+l)=l=0k1mi(xj+l+1xj+l)l=0k1mj+l(xj+l+1xj+l).
Summing both sides over i=0,1,,n1, we have
sΔsΔ.
SΔSΔ is similarly proved. ■

If we consider a sequence of finer partitions Δ,Δ,Δ,, we have 
sΔsΔsΔSΔSΔSΔ
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., SΔ, etc). By the continuity axiom of real numbers, supΔsΔ exists (supΔ means supremum over all possible partitions Δ). Similarly, infΔSΔ exists. Clearly,
supΔsΔinfΔSΔ.

Based on the above observation, we define the definite integral as follows.

Definition (Definite integral)

Let f(x) be a bounded function on the closed interval [a,b]. Let sΔ and SΔ be the lower and upper Riemann sums of f(x) with respect to the partition Δ. The function f(x) is said to be Riemann-integrable, or simply integrable, if supΔsΔ=infΔSΔ and we write
abf(x)dx=supΔsΔ(=infΔSΔ)
which we call the definite integral of f(x) on [a,b].

Example. Let us define the following function on [0,1] (called the Dirichlet function):
f(x)={1if xQ,0if xQ.
This function is not Riemann-integrable. Take any sub-interval [a,b] of any partition Δ. Since rational numbers are dense, there is always at least one rational number in [a,b]. Hence SΔ=1. Similarly, irrational numbers are dense so that sΔ=0. Therefore
supΔsΔ=0<1=infΔSΔ.

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