Riemann integral

In a branch of mathematics known as real analysis, the Riemann integral is a simple way of viewing the integral of a function on an interval as the area under the curve.

Overview

Figure 1: Illustration of Riemann integration as a converging sequence

The numbers that appear in the upper righthand corner of the animation above give the sums of the areas of the grey rectangles. As the number of rectangles increases this sum converges to the Riemann integral for the curve shown.

Figure 2

Let f(x) be a real-valued function of the interval [a,b], so that for all x, f(x) ≥ 0 (f is non-negative.) Further let S = S_f := { (x, y) | 0 ≤ y ≤ f(x) } (see Figure 2) be the region of the plane under the function f(x) and above the interval [a,b]. We are interested in measuring the area of S if that is possible, and we denote this area by ∫_a^b f(x) dx. If the variable of integration and interval of integration are understood, the notation can be simplified to ∫f.

Once we have succeeded in evaluating the integral of f for certain f which are non-negative, we can extend the integral to functions which may take negative values by linearity. Some functions have no clear Riemann integral, but especially, the interactions of limits and the Riemann integral are difficult to study. An improvement is to use the Lebesgue integral which both succeeds at integrating a broader variety of functions, as well as better describing the interactions of limits and integrals.

Historically, Riemann designed this theory first and gave some evidence for the fundamental theorem of calculus. The theory of Lebesgue integration arrived much later, when the weaknesses of the Riemann integral were better understood.

The basic idea of the Riemann integral is to use very simple and unambiguous approximations for the area of S. We find an approximate area which we are certain is less than the area of S, and we find an approximate area which we are certain is more than the area of S. If these approximations can be made arbitrarily close to one another, then we can assign an area to S.

Because of the geometric nature of the Riemann integral, it allows us to formulate many problems of nature as a problem of integration. It also provides some hints for methods of numerical integration, for evaluating definite integrals on computers to an acceptable degree of precision. However, for exact calculations for given formulae, the Riemann integral does not suggest a suitable approach.

For certain functions, the theory of antiderivatives provides exact results for definite integrals. While the Riemann integration theory justifies taking limits and provides a geometric point of view, the antiderivative theory of integration gives tools for integrating certain formulae precisely.

The fact that the seemingly disparate theories of Riemann integration and antiderivatives are essentially talking about the same subject is contained in the fundamental theorem of calculus.

Partitions of an interval and upper and lower Riemann sums

A partition of an interval a, b is a finite sequence a = x₀ < x₁ < x₂ < ... < x_n = b.

Lower and upper Riemann sums

For each subinterval x_i−1, x_i, let M_i and m_i be respectively the supremum and the infimum of the set { f(x) : x_i−1 ≤ x ≤ x_i }. Then the upper Riemann sum and the lower Rieman sum are respectively

and

.

Definition of the Riemann integral

The Riemann integral is defined to be the only number that is less than or equal to all upper sums (as the partition varies); it is both the infimum of the set of all upper sums and the supremum of the set of all lower sums.

If the supremum of the set of all lower sums differs from the infimum of the set of all upper sums, then then the function is not Riemann-integrable.

It cannot happen that the supremum is larger than the infimum (this is by our construction, as the reader may check.) However, it may happen that the supremum is less, and not equal to, the infimum. For instance, the reader may check that, for the indicating function

X_Q

where Q is the set of rational numbers in [a,b], a<b, the lower sum is 0 and the upper sum is b-a>0.

The collection of functions whose lower sum and upper sum are equal and finite is the set of Riemann integrable functions. By contrast, functions that have differing upper and lower sums are said to be non-Riemann integrable. In the context of this article, we will say integrable or non-integrable with the understanding that we are speaking of Riemann integrability.

One also checks that a step function's integral is equal to is lower and upper sums.

Results about the Riemann integral

Lemma 1: Let [a,b] be an interval. The map I:f→∫f which maps f to its integral from a to b is a linear map. That is, for any integrable functions f and g, and any real number a, I(af+g)=aI(f)+I(g).

This can be shown from first principles, from the construction of the Riemann integral.

Theorem 2: Any real-valued continuous function of the interval [a,b] is integrable.

The proof relies on the fact that any continuous function of an interval is necessarely uniformly continuous.

Corollary 3: If f is continuous everywhere in [a,b] except perhaps for finitely many points of discontinuity, and f is bounded, then f is integrable.

The boundedness requirement can not be dropped.

Theorem 4: If f_k is a sequence of integrable functions over [a,b], and if f_k converge uniformly to a function f, then f is integrable, and the integrals ∫f_k converge to ∫f.

Corollary 5: Let C(a,b) be the Banach space of continuous functions over [a,b] with the uniform norm. Then I:f→∫f is continuous. Together with Lemma 1, this says that the integral is a continuous functional of C(a,b).

The hypotheses of theorem 4 (uniform convergence on a fixed, bounded interval) are very strong. A primary failing of the Riemann integral is the difficulty we face when attempting to relax these hypotheses. In fact, the numerical sequence ∫f_k will converge to the number ∫f a lot more often than is suggested by the theorem, but it is very difficult to prove so in this setting. The correct way of getting a stronger theorem is to use the Lebesgue integral.

Another problem with the Riemann integral is that it does not extend to unbounded intervals very succesfully. If we wish to integrate a function f from -∞ to +∞, we can naively calculate

lim_n→+∞ ∫_-nⁿf(x)dx

However, certain properties (such as translation invariance, the fact that the Riemann integral does not change if we translate the integrand f) are lost. In fact, Theorem 4 becomes false for such an integral, and it becomes very difficult to use limits in conjunctions with integrals. Such an integral is called an improper integral, for it is not deemed to be a Riemann integral, strictly speaking. Again, the Lebesgue integral alleviates these difficulties.

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