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Riemann integral

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File:Integral as region under curve.svg
The integral as the area of a region under a curve.
File:Riemann integral regular.gif
A sequence of Riemann sums over a regular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function.
File:Riemann integral irregular.gif
The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero.

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In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868.<ref>The Riemann integral was introduced in Bernhard Riemann's paper "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (On the representability of a function by a trigonometric series; i.e., when can a function be represented by a trigonometric series). This paper was submitted to the University of Göttingen in 1854 as Riemann's Habilitationsschrift (qualification to become an instructor). It was published in 1868 in Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen (Proceedings of the Royal Philosophical Society at Göttingen), vol. 13, pages 87-132. (Available online here.) For Riemann's definition of his integral, see section 4, "Über den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit" (On the concept of a definite integral and the extent of its validity), pages 101–103.</ref> For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo integration.

OverviewEdit

Imagine you have a curve on a graph, and the curve stays above the x-axis between two points, a and b. The area under that curve, from a to b, is what we want to figure out. This area can be described as the set of all points (x, y) on the graph that follow these rules: a ≤ x ≤ b (the x-coordinate is between a and b) and 0 < y < f(x) (the y-coordinate is between 0 and the height of the curve f(x)). Mathematically, this region can be expressed in set-builder notation as <math display="block">S = \left \{ (x, y)\,: \,a \leq x \leq b\,,\, 0 < y < f(x) \right \}.</math>

To measure this area, we use a Riemann integral, which is written as: <math display="block">\int_a^b f(x)\,dx.</math>

This notation means “the integral of f(x) from a to b,” and it represents the exact area under the curve f(x) and above the x-axis, between x = a and x = b.

The idea behind the Riemann integral is to break the area into small, simple shapes (like rectangles), add up their areas, and then make the rectangles smaller and smaller to get a better estimate. In the end, when the rectangles are infinitely small, the sum gives the exact area, which is what the integral represents.

If the curve dips below the x-axis, the integral gives a signed area. This means the integral adds the part above the x-axis as positive and subtracts the part below the x-axis as negative. So, the result of <math>\int_a^b f(x)\,dx</math> can be positive, negative, or zero, depending on how much of the curve is above or below the x-axis.

DefinitionEdit

Partitions of an intervalEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A partition of an interval Template:Math is a finite sequence of numbers of the form <math display="block">a = x_0 < x_1 < x_2 < \dots < x_i < \dots < x_n = b</math>

Each Template:Math is called a sub-interval of the partition. The mesh or norm of a partition is defined to be the length of the longest sub-interval, that is, <math display="block">\max \left(x_{i+1}-x_i\right), \quad i \in [0,n-1].</math>

A tagged partition Template:Math of an interval Template:Math is a partition together with a choice of a sample point within each sub-interval: that is, numbers Template:Math with Template:Math for each Template:Mvar. The mesh of a tagged partition is the same as that of an ordinary partition.

Suppose that two partitions Template:Math and Template:Math are both partitions of the interval Template:Math. We say that Template:Math is a refinement of Template:Math if for each integer Template:Mvar, with Template:Math, there exists an integer Template:Math such that Template:Math and such that Template:Math for some Template:Mvar with Template:Math. That is, a tagged partition breaks up some of the sub-intervals and adds sample points where necessary, "refining" the accuracy of the partition.

We can turn the set of all tagged partitions into a directed set by saying that one tagged partition is greater than or equal to another if the former is a refinement of the latter.

Riemann sumEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Let Template:Mvar be a real-valued function defined on the interval Template:Math. The Riemann sum of Template:Mvar with respect to a tagged partition Template:Math of Template:Math is<ref>Template:Cite book</ref> <math display="block">\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math>

Each term in the sum is the product of the value of the function at a given point and the length of an interval. Consequently, each term represents the (signed) area of a rectangle with height Template:Math and width Template:Math. The Riemann sum is the (signed) area of all the rectangles.

Closely related concepts are the lower and upper Darboux sums. These are similar to Riemann sums, but the tags are replaced by the infimum and supremum (respectively) of Template:Mvar on each sub-interval: <math display="block">\begin{align} L(f, P) &= \sum_{i=0}^{n-1} \inf_{t \in [x_i, x_{i+1}]} f(t)(x_{i+1} - x_i), \\ U(f, P) &= \sum_{i=0}^{n-1} \sup_{t \in [x_i, x_{i+1}]} f(t)(x_{i+1} - x_i). \end{align}</math>

If Template:Mvar is continuous, then the lower and upper Darboux sums for an untagged partition are equal to the Riemann sum for that partition, where the tags are chosen to be the minimum or maximum (respectively) of Template:Mvar on each subinterval. (When Template:Mvar is discontinuous on a subinterval, there may not be a tag that achieves the infimum or supremum on that subinterval.) The Darboux integral, which is similar to the Riemann integral but based on Darboux sums, is equivalent to the Riemann integral.

Template:Anchor Riemann integralEdit

Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be integrable (or more specifically Riemann-integrable). The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.<ref>Template:Cite book</ref>

One important requirement is that the mesh of the partitions must become smaller and smaller, so that it has the limit zero. If this were not so, then we would not be getting a good approximation to the function on certain subintervals. In fact, this is enough to define an integral. To be specific, we say that the Riemann integral of Template:Mvar exists and equals Template:Mvar if the following condition holds:

For all Template:Math, there exists Template:Math such that for any tagged partition Template:Math and Template:Math whose mesh is less than Template:Mvar, we have <math display="block">\left| \left( \sum_{i=0}^{n-1} f(t_i) (x_{i+1}-x_i) \right) - s\right| < \varepsilon.</math>

Unfortunately, this definition is very difficult to use. It would help to develop an equivalent definition of the Riemann integral which is easier to work with. We develop this definition now, with a proof of equivalence following. Our new definition says that the Riemann integral of Template:Mvar exists and equals Template:Mvar if the following condition holds:

For all Template:Math, there exists a tagged partition Template:Math and Template:Math such that for any tagged partition Template:Math and Template:Math which is a refinement of Template:Math and Template:Math, we have <math display="block">\left| \left( \sum_{i=0}^{n-1} f(t_i) (x_{i+1}-x_i) \right) - s\right| < \varepsilon.</math>

Both of these mean that eventually, the Riemann sum of Template:Mvar with respect to any partition gets trapped close to Template:Mvar. Since this is true no matter how close we demand the sums be trapped, we say that the Riemann sums converge to Template:Mvar. These definitions are actually a special case of a more general concept, a net.

As we stated earlier, these two definitions are equivalent. In other words, Template:Mvar works in the first definition if and only if Template:Mvar works in the second definition. To show that the first definition implies the second, start with an Template:Mvar, and choose a Template:Mvar that satisfies the condition. Choose any tagged partition whose mesh is less than Template:Mvar. Its Riemann sum is within Template:Mvar of Template:Mvar, and any refinement of this partition will also have mesh less than Template:Mvar, so the Riemann sum of the refinement will also be within Template:Mvar of Template:Mvar.

To show that the second definition implies the first, it is easiest to use the Darboux integral. First, one shows that the second definition is equivalent to the definition of the Darboux integral; for this see the Darboux integral article. Now we will show that a Darboux integrable function satisfies the first definition. Fix Template:Mvar, and choose a partition Template:Math such that the lower and upper Darboux sums with respect to this partition are within Template:Math of the value Template:Mvar of the Darboux integral. Let <math display="block"> r = 2\sup_{x \in [a, b]} |f(x)|.</math>

If Template:Math, then Template:Mvar is the zero function, which is clearly both Darboux and Riemann integrable with integral zero. Therefore, we will assume that Template:Math. If Template:Math, then we choose Template:Mvar such that <math display="block">\delta < \min \left \{\frac{\varepsilon}{2r(m-1)}, \left(y_1 - y_0\right), \left(y_2 - y_1\right), \cdots, \left(y_m - y_{m-1}\right) \right \}</math>

If Template:Math, then we choose Template:Mvar to be less than one. Choose a tagged partition Template:Math and Template:Math with mesh smaller than Template:Mvar. We must show that the Riemann sum is within Template:Mvar of Template:Mvar.

To see this, choose an interval Template:Math. If this interval is contained within some Template:Math, then <math display="block"> m_j \leq f(t_i) \leq M_j</math> where Template:Mvar and Template:Mvar are respectively, the infimum and the supremum of f on Template:Math. If all intervals had this property, then this would conclude the proof, because each term in the Riemann sum would be bounded by a corresponding term in the Darboux sums, and we chose the Darboux sums to be near Template:Mvar. This is the case when Template:Math, so the proof is finished in that case.

Therefore, we may assume that Template:Math. In this case, it is possible that one of the Template:Math is not contained in any Template:Math. Instead, it may stretch across two of the intervals determined by Template:Math. (It cannot meet three intervals because Template:Mvar is assumed to be smaller than the length of any one interval.) In symbols, it may happen that <math display="block">y_j < x_i < y_{j+1} < x_{i+1} < y_{j+2}.</math>

(We may assume that all the inequalities are strict because otherwise we are in the previous case by our assumption on the length of Template:Mvar.) This can happen at most Template:Math times.

To handle this case, we will estimate the difference between the Riemann sum and the Darboux sum by subdividing the partition Template:Math at Template:Math. The term Template:Math in the Riemann sum splits into two terms: <math display="block">f\left(t_i\right)\left(x_{i+1}-x_i\right) = f\left(t_i\right)\left(x_{i+1}-y_{j+1}\right)+f\left(t_i\right)\left(y_{j+1}-x_i\right).</math>

Suppose, without loss of generality, that Template:Math. Then <math display="block">m_j \leq f(t_i) \leq M_j,</math> so this term is bounded by the corresponding term in the Darboux sum for Template:Mvar. To bound the other term, notice that <math display="block">x_{i+1}-y_{j+1} < \delta < \frac{\varepsilon}{2r(m-1)},</math>

It follows that, for some (indeed any) Template:Math, <math display="block">\left|f\left(t_i\right)-f\left(t_i^*\right)\right|\left(x_{i+1}-y_{j+1}\right) < \frac{\varepsilon}{2(m-1)}.</math>

Since this happens at most Template:Math times, the distance between the Riemann sum and a Darboux sum is at most Template:Math. Therefore, the distance between the Riemann sum and Template:Mvar is at most Template:Mvar.

ExamplesEdit

Let <math>f:[0,1]\to\R</math> be the function which takes the value 1 at every point. Any Riemann sum of Template:Mvar on Template:Math will have the value 1, therefore the Riemann integral of Template:Mvar on Template:Math is 1.

Let <math>I_{\Q}:[0,1]\to\R</math> be the indicator function of the rational numbers in Template:Math; that is, <math>I_{\Q}</math> takes the value 1 on rational numbers and 0 on irrational numbers. This function does not have a Riemann integral. To prove this, we will show how to construct tagged partitions whose Riemann sums get arbitrarily close to both zero and one.

To start, let Template:Math and Template:Math be a tagged partition (each Template:Mvar is between Template:Mvar and Template:Math). Choose Template:Math. The Template:Mvar have already been chosen, and we can't change the value of Template:Mvar at those points. But if we cut the partition into tiny pieces around each Template:Mvar, we can minimize the effect of the Template:Mvar. Then, by carefully choosing the new tags, we can make the value of the Riemann sum turn out to be within Template:Mvar of either zero or one.

Our first step is to cut up the partition. There are Template:Mvar of the Template:Mvar, and we want their total effect to be less than Template:Mvar. If we confine each of them to an interval of length less than Template:Math, then the contribution of each Template:Mvar to the Riemann sum will be at least Template:Math and at most Template:Math. This makes the total sum at least zero and at most Template:Mvar. So let Template:Mvar be a positive number less than Template:Math. If it happens that two of the Template:Mvar are within Template:Mvar of each other, choose Template:Mvar smaller. If it happens that some Template:Mvar is within Template:Mvar of some Template:Mvar, and Template:Mvar is not equal to Template:Mvar, choose Template:Mvar smaller. Since there are only finitely many Template:Mvar and Template:Mvar, we can always choose Template:Mvar sufficiently small.

Now we add two cuts to the partition for each Template:Mvar. One of the cuts will be at Template:Math, and the other will be at Template:Math. If one of these leaves the interval [0, 1], then we leave it out. Template:Mvar will be the tag corresponding to the subinterval <math display="block">\left [t_i - \frac{\delta}{2}, t_i + \frac{\delta}{2} \right ].</math>

If Template:Mvar is directly on top of one of the Template:Mvar, then we let Template:Mvar be the tag for both intervals: <math display="block">\left [t_i - \frac{\delta}{2}, x_j \right ], \quad\text{and}\quad \left [x_j,t_i + \frac{\delta}{2} \right ].</math>

We still have to choose tags for the other subintervals. We will choose them in two different ways. The first way is to always choose a rational point, so that the Riemann sum is as large as possible. This will make the value of the Riemann sum at least Template:Math. The second way is to always choose an irrational point, so that the Riemann sum is as small as possible. This will make the value of the Riemann sum at most Template:Mvar.

Since we started from an arbitrary partition and ended up as close as we wanted to either zero or one, it is false to say that we are eventually trapped near some number Template:Mvar, so this function is not Riemann integrable. However, it is Lebesgue integrable. In the Lebesgue sense its integral is zero, since the function is zero almost everywhere. But this is a fact that is beyond the reach of the Riemann integral.

There are even worse examples. <math>I_{\Q}</math> is equivalent (that is, equal almost everywhere) to a Riemann integrable function, but there are non-Riemann integrable bounded functions which are not equivalent to any Riemann integrable function. For example, let Template:Mvar be the Smith–Volterra–Cantor set, and let Template:Math be its indicator function. Because Template:Mvar is not Jordan measurable, Template:Math is not Riemann integrable. Moreover, no function Template:Mvar equivalent to Template:Math is Riemann integrable: Template:Mvar, like Template:Math, must be zero on a dense set, so as in the previous example, any Riemann sum of Template:Mvar has a refinement which is within Template:Mvar of 0 for any positive number Template:Mvar. But if the Riemann integral of Template:Mvar exists, then it must equal the Lebesgue integral of Template:Math, which is Template:Math. Therefore, Template:Mvar is not Riemann integrable.

Similar conceptsEdit

It is popular to define the Riemann integral as the Darboux integral. This is because the Darboux integral is technically simpler and because a function is Riemann-integrable if and only if it is Darboux-integrable.

Some calculus books do not use general tagged partitions, but limit themselves to specific types of tagged partitions. If the type of partition is limited too much, some non-integrable functions may appear to be integrable.

One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, Template:Math for all Template:Mvar, and in a right-hand Riemann sum, Template:Math for all Template:Mvar. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each Template:Mvar. In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.

Another popular restriction is the use of regular subdivisions of an interval. For example, the Template:Mvarth regular subdivision of Template:Math consists of the intervals <math display="block">\left [0, \frac{1}{n} \right], \left [\frac{1}{n}, \frac{2}{n} \right], \ldots, \left[\frac{n-1}{n}, 1 \right].</math>

Again, alone this restriction does not impose a problem, but the reasoning required to see this fact is more difficult than in the case of left-hand and right-hand Riemann sums.

However, combining these restrictions, so that one uses only left-hand or right-hand Riemann sums on regularly divided intervals, is dangerous. If a function is known in advance to be Riemann integrable, then this technique will give the correct value of the integral. But under these conditions the indicator function <math>I_{\Q}</math> will appear to be integrable on Template:Math with integral equal to one: Every endpoint of every subinterval will be a rational number, so the function will always be evaluated at rational numbers, and hence it will appear to always equal one. The problem with this definition becomes apparent when we try to split the integral into two pieces. The following equation ought to hold: <math display="block">\int_0^{\sqrt{2}-1} I_\Q(x) \,dx + \int_{\sqrt{2}-1}^1 I_\Q(x) \,dx = \int_0^1 I_\Q(x) \,dx.</math>

If we use regular subdivisions and left-hand or right-hand Riemann sums, then the two terms on the left are equal to zero, since every endpoint except 0 and 1 will be irrational, but as we have seen the term on the right will equal 1.

As defined above, the Riemann integral avoids this problem by refusing to integrate <math>I_{\Q}.</math> The Lebesgue integral is defined in such a way that all these integrals are 0.

PropertiesEdit

LinearityEdit

The Riemann integral is a linear transformation; that is, if Template:Mvar and Template:Mvar are Riemann-integrable on Template:Math and Template:Mvar and Template:Mvar are constants, then <math display="block">\int_{a}^{b} (\alpha f(x) + \beta g(x))\,dx = \alpha \int_{a}^{b}f(x)\,dx + \beta \int_{a}^{b}g(x)\,dx. </math>

Because the Riemann integral of a function is a number, this makes the Riemann integral a linear functional on the vector space of Riemann-integrable functions.

IntegrabilityEdit

A bounded function on a compact interval Template:Math is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is the Template:Visible anchor (of characterization of the Riemann integrable functions). It has been proven independently by Giuseppe Vitali and by Henri Lebesgue in 1907, and uses the notion of measure zero, but makes use of neither Lebesgue's general measure or integral.

The integrability condition can be proven in various ways,<ref name="apostol169">Template:Harvnb</ref><ref>Template:Cite journal</ref><ref>Basic real analysis, by Houshang H. Sohrab, section 7.3, Sets of Measure Zero and Lebesgue’s Integrability Condition, pp. 264–271</ref><ref>Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177</ref> one of which is sketched below.

In particular, any set that is at most countable has Lebesgue measure zero, and thus a bounded function (on a compact interval) with only finitely or countably many discontinuities is Riemann integrable. Another sufficient criterion to Riemann integrability over Template:Math, but which does not involve the concept of measure, is the existence of a right-hand (or left-hand) limit at every point in Template:Math (or Template:Math).<ref>Template:Cite journal</ref>

An indicator function of a bounded set is Riemann-integrable if and only if the set is Jordan measurable. The Riemann integral can be interpreted measure-theoretically as the integral with respect to the Jordan measure.

If a real-valued function is monotone on the interval Template:Math it is Riemann integrable, since its set of discontinuities is at most countable, and therefore of Lebesgue measure zero. If a real-valued function on Template:Math is Riemann integrable, it is Lebesgue integrable. That is, Riemann-integrability is a stronger (meaning more difficult to satisfy) condition than Lebesgue-integrability. The converse does not hold; not all Lebesgue-integrable functions are Riemann integrable.

The Lebesgue–Vitali theorem does not imply that all type of discontinuities have the same weight on the obstruction that a real-valued bounded function be Riemann integrable on Template:Math. In fact, certain discontinuities have absolutely no role on the Riemann integrability of the function—a consequence of the classification of the discontinuities of a function.Template:Citation needed

If Template:Math is a uniformly convergent sequence on Template:Math with limit Template:Mvar, then Riemann integrability of all Template:Math implies Riemann integrability of Template:Mvar, and <math display="block"> \int_{a}^{b} f\, dx = \int_a^b{\lim_{n \to \infty}{f_n}\, dx} = \lim_{n \to \infty} \int_{a}^{b} f_n\, dx.</math>

However, the Lebesgue monotone convergence theorem (on a monotone pointwise limit) does not hold for Riemann integrals. Thus, in Riemann integration, taking limits under the integral sign is far more difficult to logically justify than in Lebesgue integration.<ref>Template:Cite journal</ref>

GeneralizationsEdit

It is easy to extend the Riemann integral to functions with values in the Euclidean vector space <math>\R^n</math> for any Template:Mvar. The integral is defined component-wise; in other words, if Template:Math then <math display="block">\int\mathbf{f} = \left(\int f_1,\,\dots, \int f_n\right).</math>

In particular, since the complex numbers are a real vector space, this allows the integration of complex valued functions.

The Riemann integral is only defined on bounded intervals, and it does not extend well to unbounded intervals. The simplest possible extension is to define such an integral as a limit, in other words, as an improper integral: <math display="block">\int_{-\infty}^\infty f(x)\,dx = \lim_{a \to -\infty \atop b \to \infty}\int_a^b f(x)\,dx.</math>

This definition carries with it some subtleties, such as the fact that it is not always equivalent to compute the Cauchy principal value <math display="block">\lim_{a\to\infty} \int_{-a}^a f(x)\,dx.</math>

For example, consider the sign function Template:Math which is 0 at Template:Math, 1 for Template:Math, and −1 for Template:Math. By symmetry, <math display="block">\int_{-a}^a f(x)\,dx = 0</math> always, regardless of Template:Mvar. But there are many ways for the interval of integration to expand to fill the real line, and other ways can produce different results; in other words, the multivariate limit does not always exist. We can compute <math display="block">\begin{align} \int_{-a}^{2a} f(x)\,dx &= a, \\ \int_{-2a}^a f(x)\,dx &= -a. \end{align}</math>

In general, this improper Riemann integral is undefined. Even standardizing a way for the interval to approach the real line does not work because it leads to disturbingly counterintuitive results. If we agree (for instance) that the improper integral should always be <math display="block">\lim_{a\to\infty} \int_{-a}^a f(x)\,dx,</math> then the integral of the translation Template:Math is −2, so this definition is not invariant under shifts, a highly undesirable property. In fact, not only does this function not have an improper Riemann integral, its Lebesgue integral is also undefined (it equals Template:Math).

Unfortunately, the improper Riemann integral is not powerful enough. The most severe problem is that there are no widely applicable theorems for commuting improper Riemann integrals with limits of functions. In applications such as Fourier series it is important to be able to approximate the integral of a function using integrals of approximations to the function. For proper Riemann integrals, a standard theorem states that if Template:Math is a sequence of functions that converge uniformly to Template:Mvar on a compact set Template:Math, then <math display="block">\lim_{n\to\infty} \int_a^b f_n(x)\,dx = \int_a^b f(x)\,dx.</math>

On non-compact intervals such as the real line, this is false. For example, take Template:Math to be Template:Math on Template:Math and zero elsewhere. For all Template:Mvar we have: <math display="block">\int_{-\infty}^\infty f_n\,dx = 1.</math>

The sequence Template:Math converges uniformly to the zero function, and clearly the integral of the zero function is zero. Consequently, <math display="block">\int_{-\infty}^\infty f\,dx \neq \lim_{n\to\infty}\int_{-\infty}^\infty f_n\,dx.</math>

This demonstrates that for integrals on unbounded intervals, uniform convergence of a function is not strong enough to allow passing a limit through an integral sign. This makes the Riemann integral unworkable in applications (even though the Riemann integral assigns both sides the correct value), because there is no other general criterion for exchanging a limit and a Riemann integral, and without such a criterion it is difficult to approximate integrals by approximating their integrands.

A better route is to abandon the Riemann integral for the Lebesgue integral. The definition of the Lebesgue integral is not obviously a generalization of the Riemann integral, but it is not hard to prove that every Riemann-integrable function is Lebesgue-integrable and that the values of the two integrals agree whenever they are both defined. Moreover, a function Template:Mvar defined on a bounded interval is Riemann-integrable if and only if it is bounded and the set of points where Template:Mvar is discontinuous has Lebesgue measure zero.

An integral which is in fact a direct generalization of the Riemann integral is the Henstock–Kurzweil integral.

Another way of generalizing the Riemann integral is to replace the factors Template:Math in the definition of a Riemann sum by something else; roughly speaking, this gives the interval of integration a different notion of length. This is the approach taken by the Riemann–Stieltjes integral.

In multivariable calculus, the Riemann integrals for functions from <math>\R^n\to\R</math> are multiple integrals.

Comparison with other theories of integrationEdit

The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral, though the latter does not have a satisfactory treatment of improper integrals. The gauge integral is a generalisation of the Lebesgue integral that is at the same time closer to the Riemann integral. These more general theories allow for the integration of more "jagged" or "highly oscillating" functions whose Riemann integral does not exist; but the theories give the same value as the Riemann integral when it does exist.

In educational settings, the Darboux integral offers a simpler definition that is easier to work with; it can be used to introduce the Riemann integral. The Darboux integral is defined whenever the Riemann integral is, and always gives the same result. Conversely, the gauge integral is a simple but more powerful generalization of the Riemann integral and has led some educators to advocate that it should replace the Riemann integral in introductory calculus courses.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

See alsoEdit

NotesEdit

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ReferencesEdit

  • Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. Template:Isbn.
  • Template:Citation

External linksEdit

Template:Integral Template:Bernhard Riemann Template:Authority control

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