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If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits.
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. The Riemann-Stieltjes integral is a generalization of the Riemann integral that addresses some of
the problems noted below in the Riemann integral. Another formulation of integration that removes more problems is the Lebesgue integral.
Overview
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 = Sf := { (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
∫ab 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 <
... < xn = b.
Lower and upper Riemann sums
For each subinterval [xi−1, xi], let
Mi and mi be respectively the supremum and the infimum of the set { f(x) :
xi−1 ≤ x ≤ xi }. 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
- XQ
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.
One may formulate an equivalent definition for Riemann integrability involving an epsilon-partition argument. It is left as an
exercise for the reader to show that the following criterion is equivalent to the above criterion for integrability. A function
f is said to be Riemann integrable on [a,b] if for every ε > 0 there exists a partition
P, such that U(f,P) − L(f,P) are the upper and lower sums of
f with respect to P
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 fk is a sequence of integrable functions over [a,b],
and if fk converge uniformly to a function f, then f is integrable, and the integrals ∫fk
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
∫fk 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 successfully. If we wish to
integrate a function f from −∞ to +∞, we can naively calculate
-
(see improper integral). 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.
See also
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