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Definition: Integral |
IntegralAdjective1. Existing as an essential constituent or characteristic; "the Ptolemaic system with its built-in concept of periodicity"; "a constitutional inability to tell the truth". 2. Constituting the undiminished entirety; lacking nothing essential especially not damaged; "a local motion keepeth bodies integral"- Bacon; "was able to keep the collection entire during his lifetime"; "fought to keep the union intact". Noun1. The result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x). Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "integral" was first used: 1471. (references) |
| Domain | Definition |
Aerospace | 1. Of or pertaining to an integer.2. Serving to form a whole or part of a whole, as an integral tank. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
In differential geometry, a differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of a manifold. At any point p on a manifold, a k-form gives a multilinear map from the k-th cartesian power of the tangent space at p to R.For example, the differential of a smooth function on a manifold (a 0-form) is a 1-form.
1-forms are a particularly useful basic concept in the coordinate-free treatment of tensors. In this context, they can be defined as real-valued linear functions of vectors, and they can be seen to create a dual space with regard to the vector space of the vectors they are defined over. An older name for 1-forms in this context is "covariant vectors".
Integration of forms
Differential forms of degree k are integrated over k dimensional chainss. If , this is just evaluation of functions at points. Other values of correspond to line integrals, surface integrals, volume integrals etc.
See also Stokes' theorem.
Operations on forms
The set of all k-forms on a manifold is a vector space. Furthermore, there are two other operations: wedge product and exterior derivative. d2=0, see de Rham cohomology for more details.
The fundamental relationship between the exterior derivative and integration is given by the general Stokes' theorem, which also provides the duality between de Rham cohomology and the homology of chains.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Differential form."
(From Wikipedia, the free Encyclopedia)
The integers consist of the natural numbers (0, 1, 2, ...) and their negatives (-1, -2, -3, ...; -0 is equal to 0 and therefore not included as a separate integer). The set of all integers is usually denoted by Z (or Z in blackboard bold, ), which stands for Zahlen (German for "numbers").
Integers can be added and subtracted, multiplied, and compared. Introducing the negative integers makes it possible to solve all equations of the form
(where a and b are constant natural numbers) for the unknown x; if x is constrained to the natural numbers, only some of these equations are solvable.
- a + x = b
Mathematicians express the fact that all the usual laws of arithmetic are valid in the integers by saying that (Z, +, *) is a commutative ring.
Z is a totally ordered set without upper or lower bound. The ordering of Z is given by
We call an integer positive if it is greater than zero; zero itself is not considered to be positive. The order is compatible with the algebraic operations in the following way:
- ... < -2 < -1 < 0 < 1 < 2 < ...
Like the natural numbers, the integers form a countably infinite set.
- if a < b and c < d, then a + c < b + d
- if a < b and 0 < c, then ac < bc
The integers do not form a field since for instance there is no integer x such that 2x = 1. The smallest field containing the integers is the rational numbers.
An important property of the integers is division with remainder: given two integers a and b with b≠0, we can always find integers q and r such that
and such that 0 <= r < |b| (see absolute value). q is called the quotient and r is called the remainder resulting from division of a by b. The numbers q and r are uniquely determined by a and b. This shows that the greatest common divisor of two integers can always be written as a sum of multiples of the two numbers, and makes the Euclidean algorithm for computing greatest common divisors possible.
- a = b q + r
All of this can be abbreviated by saying that Z is a Euclidean domain. This implies that Z is a principal ideal domain and that whole numbers can be written as products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.
The branch of mathematics which studies the integers is called number theory.
An integer is often one of the primitive datatypes in computer languages. Note, however, that a computer can only represent a subset of all mathematical integers, given that computers are finite machines. Integer datatypes are typically implemented using a fixed number of bits, and even variable-length representations eventually run out of storage space when trying to represent especially large numbers. See integer (computer science) for more detailed discussion.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Integer."
(From Wikipedia, the free Encyclopedia)
For non-mathematical meanings of "Integral", see integration (non-mathematical). It is recommended that the reader be familiar with algebra, derivatives, functions, and limitss.
In mathematics, the term "integral" has two unrelated meanings; one relating to integers, the other relating to integral calculus.
Integral Values
A real number is "integral" if it is an integer. The integral value, of a real number x, is defined as the largest integer which is less than, or equal to, x; this is often denoted by ; known as the "floor function".
Integral Calculus
In calculus, the integral, of a function, is the size of the area bounded by the x-axis and the graph of a function, f(x); negative areas are possible. Integrals are calculated by integration, which is a so-called "accumulation process" (see below).
Let f(x) be a function of the interval [a,b] into the real numbers. For simplicity, assume that this function is non-negative (it takes no negative values.) The set S=Sf:={(x,y)|0≤y≤f(x)} is the region of the plane between f and the x axis. Measuring the "area" of S is desirable, and this area is denoted by ∫f, and it is the (definite) integral of f.
Improper and Trigonometric Integrals
If either the interval of integration, or the range of the function, is infinite; the integral is an "improper integral". Integrals which involve trigonometric functions, are trigonometric integrals. Some integrals can be evaluated via trigonometric substitution.
Means of Integration
The following pages discuss means of integrating various functions:
- disk integration
- list of integrals
- shell integration
- trigonometric integration
Riemann and Lebesgue Integrals
One should examine the articles on Riemann and Lebesgue integrals. The concept of Riemann integration was developed first, and Lebesgue integrals were developed to deal with pathological cases for which the Riemann integral was not defined. If a function is Riemann integrable, then it is also Lebesgue integrable, and the two integrals coincide.The antiderivative approach occurs when we seek to find a function F(x) whose derivative F(x) is some given function f(x). This approach is motivated by calculus, and is the main method used for calculating the area under the curve as described in the preceding paragraph, for functions given by formulae.
Functions which have antiderivatives are also Riemann integrable (and hence Lebesgue integrable.) The nonobvious theorem that states that the two approaches ("area under the curve" and "antiderivative") are in some sense the same is the fundamental theorem of calculus
(And the relationships works in reverse; the Radon-Nikodym derivative can be pulled out of the measure machinery underlying Lebesgue integrals.)
The nuance between Riemann and Lebesgue integration
Both the Riemann and the Lebesgue integral are approaches to integration which seek to measure the area under the curve, and the overall schema in both cases is the same.
First, we select a family of elementary functions, for which we have an obvious way of measuring the area under the curve. In the case of the Riemann integral, this choice is so that the area under the curve can be regarded as a finite union of rectangles, and the functions are called step functions. For the Lebesgue integral, "rectangle" is replaced by something more sophisticated, and the resulting functions are called simple functions.
Then we try to impose monotonicity. If 0≤f≤g (and hence Sf is a subset of Sg) then we should have that ∫f≤∫g. With this monotonicity requirement, for an arbitrary non-negative function f, we can approximate its area from below using a carefully chosen elementary function s (in the case of Riemann integration, a step function, and in the case of Lebesgue integration, a simple function.) We choose s so that s≤f but s is very close to f. The area under s is a lower bound for the integral of f, and it is called a lower sum. In the case of the Riemann integral, we also produce upper sums in a similar fashion: we choose step functions, say s, so that s≥f but s is very close to f, and we regard such an upper sum as an upper bound for the area under f. The Lebesgue theory does not use upper sums.
Lastly, a limit-taking step is taken to make the elementary functions approach f more and more closely, and an area is obtained for some functions f. The functions which we can integrate are said to be integrable. However, the differences begin here; the Riemann theory was simpler thus far, but its simplicity results in a more limited set of integrable functions than the Lebesgue theory. In addition, the interaction between limits and the integral are more difficult to describe in the Riemann setting.
Other integrals
Although the Riemann and Lebesgue integrals are the most important ones, a number of others exist, including but not limited to:
- the Darboux integral, a variation of the Riemann integral
- the Denjoy integral, an extension of both the Riemann and Lebesgue integrals
- the Euler integral
- the Haar integral
- the Henstock-Kurzweil integral, an extension of both the Riemann and Lebesgue integrals (also called HK-integral)
- the Henstock-Kurzweil-Stieltjes integral (also called HK-Stieltjes integral)
- the Lebesgue-Stieltjes integral (also called Lebesgue-Radon integral)
- the Perron integral, which is equivalent to the restricted Denjoy integral
- the Stieltjes integral, an extension of the Riemann integral (also called Riemann-Stieltjes integral)
See also: Calculus, List of integrals
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Integral."
(From Wikipedia, the free Encyclopedia)
''This is a disambiguation page; if one followed a link here, one might want to go back and adjust that link. Integration is constructing an object, a theory, etc. from separate more limited parts. The result is something composite or integral.Integration can be:
See also:
- the process of ending racial segregation in USA, racial integration, also called desegregation
- a mathematical technique, mathematical integration, see integral
- Very-large-scale integration was a step in semiconductor technology
- Computer telephony integration, as a step in telephone technology
- an economic process; see: horizontal integration and vertical integration
- Sensory integration dysfunction, a common symptom of Autism
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Integration."
(From Wikipedia, the free Encyclopedia)
If you are having difficulty understanding this article, you might want to learn about Riemann integrals and the theory of limits first.
In the mathematical branch of real analysis, Lebesgue integration is a framework for extending the notion of integral as the area under the curve to a large class of functions whose domain may not even be in R.
In mathematics, integration is the process of calculating the area Sf under the graph of a function f.
The Lebesgue approach is not the most elementary area-based integration theory; that distinction goes to the Riemann integral. The principal advantage of the Lebesgue theory over the Riemann theory is the ease with which limit theorems are proved. Such theorems are needed in the study of Fourier series, Fourier transforms, and elsewhere. It is often also mentioned that a much broader class of functions can be integrated.
Discussion
Here we discuss the failures of the Riemann integral and the improvements offered by the Lebesgue integral. We presume a working understanding of the Riemann integral. It is very difficult to explain the subtle differences between these two integrals unless the reader is familiar with the basic working of the Riemann integral.
The Riemann integral approaches the problem of calculating surface area directly: one covers the target surface area with tiny rectangles. The sum of the areas of the rectangles is an approximation of the area of the target surface. Because it is so simple, it is easier to understand.
With the advent of Fourier series, there arose the need to exchange summation and integral signs much more often. However, the conditions under which ∑k∫fk and ∫∑kfk are equal proved quite elusive in the Riemann framework. It may come as a surprise to the casual reader that these two quantities may not be equal, so an example helps:
However, it was clear from experience that in many very useful situations, the sum and the integral did commute. It was very important to be able to describe which conditions enabled the exchange of the sum and integral signs. Unfortunately, the Riemann integral is poorly equipped to deal with this question; its main useful convergence theorem being the uniform convergence theorem: if fk are Riemann-integrable functions of [a,b] converging uniformly to f, then ∫fk converges to ∫f. Since Fourier series rarely converge uniformly, this theorem is clearly insufficient.
- Let fk(x) be 1 on (k,k+1] and -1 on (k+1,k+2] and 0 everywhere else.
- Then, ∑k=1∞fk(x)=f(x) where f(x) is zero everywhere except on (1,2] where it is 1. Hence, ∫∑fk=∫f=1.
- However, ∫fk=0 for every k, hence ∑∫fk=0.
Technical difficulties with the Riemann integral
Here we list further technical problems with the Riemann integral. These are actually intimately linked with the limit taking difficulty discussed above.
- If H(x) is a function of [0,1] which is 0 everywhere, except that it is 1 on the rational numbers (see [nowhere continuous]), then it is not Riemann integrable. This is because, in the calculation of its upper sum, any rectangle used will have height 1 (because all rectangles contain rational points) and in the lower sum, any rectangle used will have height 0 (because all rectangles contain irrational points.) Hence the lower sum is 0 and the upper sum is 1.
- This means that the monotone convergence theorem does not hold. The monotone convergence theorem would say that if fk(x) is a sequence of non-negative functions increasing monotonously in k to f(x), then the integrals of ∫fk(x)dx should converge to ∫f(x)dx. To see why this is so, let ak be an enumeration of all the rational numbers in [0,1] (they are countable so we're ok.) Then let gk be the function which is 1 on ak and 0 everywhere else. Lastly let fk=g1+g2+...+gk. Then fk is zero everywhere except on a finite set of points, hence its Riemann integral is zero. The sequence fk is also clearly non-negative and monotonously increasing to H(x), but H(x) isn't Riemann integrable.
- The Riemann integral can only integrate functions on an interval. The simplest extension is to define ∫-∞∞f(x) dx by the limit of ∫-aaf(x) dx as a goes to +∞. However, this breaks translation invariance: if f and g are zero outside some interval [a,b] and are Riemann integrable, and if f(x) = g(x+y) for some y, then ∫f=∫g. However, with this definition of the [improper integral], the functions f(x)=(1 if x>0, -1 otherwise) and g(x)=(1 if x>1, -1 otherwise) are translations of one another, but their improper integrals are different. (∫f=0 but ∫g=-2.)
Towards a better integration theory
The solution, as it turns out, is to study an even simpler problem first. The observation is that, if we have a notion of length, we can turn it into a notion of area. Instead of measuring the area of a surface in the plane, we turn our attention to measuring the length of subsets of the real line. One obvious requirement is that an interval [a,b] should have a length of b-a. What other demands we should put on the notion of length is less clear, and much effort was put into obtaining a useful definition. In fact, the term length was first used, but its construction was misguided, and a later, more useful construction is in use today; it is called the measure.
Measure theory enables us to calculate the length of subsets of the real line. It also fully classifies which sets have a length, and which sets do not have a reasonable notion of length. By spending the extra effort into calculating lengths carefully, we now have a more solid foundation to work with.
Of course, the Riemann integral uses the notion of length anonymously. Indeed, the element of calculation for the Riemann integral is the rectangle [a,b]x[c,d], whose area is calculated to be (b-a)(c-d). Obviously the numbers b-a and c-d are meant to be the lengths of [a,b] and [c,d]. However, we can now augment the Riemann integral. Indeed, Riemann could only use rectangles because he could only measure intervals. Equipped with a measure μ, we can calculate the length of sets much more interesting than intervals. So, if X and Y are μ-measurable, we can easily define the area of the cartesian product XxY to be μ(X)μ(Y). This definition clearly generalizes Riemann's notion of area of a rectangle. In the context of Lebesgue integration, sets such as XxY are sometimes called rectangles, even though they are far more complicated than the quadrilaterals of the same name.
With the ability to measure the area of more complex rectangles, we can attempt to integrate more complex functions. One crucial, but nonobvious step, was to drop the notion of upper sum. While upper sums work just fine for bounded functions of bounded intervals, there is a clear problem for unbounded functions, or functions which are supported by all of the real line. For instance, the function f(x)=1/x2 for x>1 would necessarely have an infinite upper sum, however it should be clear to any calculus student that this function has a finite integral.
Dropping the upper sum robs us of our main way of checking for integrability of functions. It isn't obvious how to decide on the integrability of functions while maintaining a consistent theory. It is very fortunate that a simple (if technical) definition is available.
The resulting theory of integration is much more accurate in describing limit taking processes. Many of the original questions posed by Fourier series (about swapping the integral and summation signs) are answerable using one or another of the various Lebesgue integral limit theorems (the main ones are monotone convergence, dominated convergence and Fatou's lemma; see below.)
Formal construction
Let μ be a (non-negative) measure on a sigma-algebra X over a set E. (In real analysis, E will typically be Euclidean n-space Rn or some Lebesgue measurable subset of it, X will be the sigma-algebra of all Lebesgue measurable subsets of E, and μ will be the Lebesgue measure. In probability and statistics, μ will be a probability measure on a probability space E.) We build up an integral for real-valued functions defined on E as follows.
Fix a set S in X and let f be the function on E whose value is 0 outside of S and 1 inside of S (i.e., f(x) = 1 if x is in S, otherwise f(x) = 0.) This is called the indicating or characteristic function of S and is denoted 1S.
To assign a value to ∫1S consistent with the given measure μ, the only reasonable choice is to set:
We extend by linearity to the linear span of indicating functions:
where the sum is finite and the coefficients ak are real numbers. Such a finite linear combination of indicating functions is called a simple function. Note that a simple function can be written in many ways as a linear combination of characteristic functions, but the integral will always be the same.
Now the difficulties begin as we attempt to take limits so that we can integrate more general functions. It turns out that the following process works and is most fruitful.
Let f be a non-negative function supported on the set E (we allow it to attain the value +∞, in other words, f takes values in the extended real number line.) We define ∫f to be the supremum of ∫s where s varies over all simple functions which are under f (that is, s(x) ≤ f(x) for all x.) This is analogous to the lower sums of Riemann. However, we will not build an upper sum, and this fact is important in getting a more general class of integrable functions. One can be more explicit and mention the measure and domain of integration:
There is the question of whether this definition makes sense (do simple function or indicating function keep the same integral?) There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is not so hard to prove that the answer to both questions is yes.
We have defined ∫f for any non-negative function on E; however for some functions ∫f will be infinite. Furthermore, desirable additive and limit properties of the integral are not satisfied, unless we require that all our functions are measurable, meaning that the pre-image of any interval is in X. We will make this assumption from now on.
To handle signed functions, we need a few more definitions. If f is a function of the measurable set E to the reals (including ± ∞), then we can write f = g - h where g(x) = (f(x) if f(x)>0, 0 otherwise) and h(x) = (-f(x) if f(x) < 0, 0 otherwise). Note that both g and h are non-negative functions. Also note that |f| = g + h. If ∫|f| is finite, then f is called Lebesgue integrable. In this case, both ∫g and ∫h are finite, and it makes sense to define ∫f by ∫g - ∫h. It turns out that this definition is the correct one. Complex valued functions can be similarly integrated, by considering the real part and the imaginary part separately.
Theorems
Every reasonable notion of integral needs to be linear and monotone, and the Lebesgue integral is: if f and g are integrable functions and a and b are real numbers, then af + bg is integrable and ∫(af + bg) = a∫f + b∫g; if f ≤ g, then ∫f ≤ ∫g.
Two functions which only differ on a set of μ-measure zero have the same integral, or more precisely: if μ({x : f(x) ≠ g(x)}) = 0, then f is integrable if and only if g is, and in this case ∫ f = ∫ g.
One of the most important advantages that the Lebesgue integral carries over the Riemann integral is the ease with which we can perform limit processes. Three theorems are key here.
The monotone convergence theorem states that if fk is a sequence of non-negative measurable functions such that fk(x) ≤ fk+1(x) for all k, and if f = lim fk, then ∫fk converges to ∫f as k goes to infinity. (Note: ∫f may be infinite here.)
Fatou's lemma states that if fk is a sequence of non-negative measurable functions and if f = liminf fk, then ∫f ≤ liminf ∫fk. (Again, ∫f may be infinite.)
The dominated convergence theorem states that if fk is a sequence of measurable functions with pointwise limit f, and if there is an integrable function g such that |fk| ≤ g for all k, then f is integrable and ∫fk converges to ∫f.
Equivalent formulations
If f is non-negative, then ∫fdμ is precisely the area under the curve as measured by the product measure μxλ where λ is the Lebesgue measure for R.
One can also circumvent measure theory entirely. For any continuous function f of compact support, the Riemann integral gives a perfectly good answer. Then we use functional analysis. Let Cc be the space of all real-valued compactly supported continuous functions of R. Define a norm on Cc by
Then Cc is a normed vector space (and in particular, it is a metric space.) All metric spaces have completions, so let L1 be its completion. This space is isomorphic to the space of Lebesgue integrable functions (modulo sets of measure zero). Furthermore, the Riemann integral ∫ defines a continuous functional on Cc which is dense in L1 hence ∫ has a unique extension to all of L1. This integral is precisely the Lebesgue integral.
- ||f||=∫|f(x)|
In this formulation, the limit taking theorems are hard to prove. However, in more general cases (such as when the functions, or perhaps the measures, take values in a large vector space instead of Rn) this approach is a fast way of obtaining an integral.
See also: null set, Henri Lebesgue, integration, measure, sigma-algebra, Lebesgue measure
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Lebesgue integration."
| The following table is compiled from various sources, across various languages. When English abbreviations or acronyms come from a non-English source, this is noted. | |||
| Entry | Source | Expression | Field |
INTEGRAL | English | International Gamma X-Ray Astrophysics Laboratory | Transportation |
Source: compiled by the editor, based on several corpora (additional references). | |||
Synonyms: IntegralSynonyms: built-in (adj), constitutional (adj), entire (adj), inbuilt (adj), inherent (adj), intact (adj). (additional references) |
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Component | Noun: component; component part, integral part, integrant part; element, ingredient, leaven; part and parcel; contents; appurtenance; feature; member; (part); personnel. |
Number | Proportional, exponential, logarithmic, logometric, differential, fluxional, integral,proportional, exponential, logarithmic, logometric, differential, fluxional, integral, totitive. |
Differential, integral, fluxion, fluent. | |
Numeration | Arithmetic, analysis, algebra, geometry, analytical geometry, fluxions; differential calculus, integral calculus, infinitesimal calculus; calculus of differences. |
Whole | Adjective: whole, total, integral, entire; complete; one, individual. |
| Source: adapted from Roget's Thesaurus. | |
Crosswords: Integral |
| English words defined with "integral": accidental, adventitious ♦ build in ♦ consolidation ♦ definite integral ♦ embedded ♦ incidental, indefinite integral, Index of a logarithm, infinitesimal calculus, Integral unit, integrally, Integrant, integrate, integration ♦ Morphotic ♦ nonessential ♦ operator ♦ power series, prime number ♦ Quantic ♦ strategic ♦ Whole snipe. (references) |
| Specialty definitions using "integral": Fourier integral ♦ integral key, integral pilot ♦ probability integral, proportional plus integral control action, proportional plus integral plus derivative control action. (references) |
| Non-English Usage: "Integral" is also a word in the following languages with English translations in parentheses. Albanian (integral), German (definite, integral), Indonesian (integral), Norwegian (integral), Portuguese (integral, integrate, unabridged, whole), Romanian (integral, integrate, whole), Serbo-Croatian (integral), Spanish (integral, integrate, uncut), Swedish (integral), Turkish (integral). |
| Domain | Title | ||
References |
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Books |
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Periodicals |
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Theater & Movies | |||
Music |
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High Tech |
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Consumer Goods | |||
Source: compiled by the editor from various references; see credits. | |||
| Thumbnail | Description & Credit | Thumbnail | Description & Credit |
 | Scientists have identified a protein, named autocrine motility factor (AMF), that causes cancer cells to grow "arms" or pseudopodia, enabling them to migrate to other parts of the body. Locomotion is integral to the entire process of metastasis. Credit: Susan Arnold (photographer). |  | The Air Force is embarking on a "cultural shift" that will bolster its recruiting force and make recruiting duty an integral part of nearly every noncommissioned officer's career path by ending its all-volunteer recruiter system and adopting a s. |
 | Irrigation canals are an integral part of irrigation water distribution systems across southern Idaho. Credit: Howard Johnson. |  | This diversion structure is an integral part of an irrigation system in Lemhi County. Credit: Bob Minton. |
 | Women are an integral part of fire crews. Credit: USDA. | ||
Source: pictures compiled by the editor from various references; see picture credits. | |||
| Author | Date | Quotation |
Treaty of Versailles | 1919 | This Annex shall be considered as an integral part of the present Treaty, and Germany declares her adherence to it. (reference) |
Source: compiled by the editor from various references. | ||
| Title | Author | Quote |
A Grief Observed | C.S. Lewis | If, as I can't help suspecting, the dead also feel the pains of separation (and this may be one of their purgatorial sufferings), then for both lovers, and for all pairs of lovers without exception, bereavement is a universal and integral part of our experience of love |
Source: compiled by the editor from various references. | ||
| Subject | Topic | Quote |
Health | Chiropractors use manipulative therapy as an integral treatment tool. (references) | |
Behavioral interventions can play an integral role in nicotine addiction treatment. (references) | ||
The cost to the patient should be considered as an integral part of adjunctive therapy. (references) | ||
Business | Convenience stores have become an integral part of the young people's life in big cities. (references) | |
While public transportation is quick and efficient, car ownership is an integral part of Singaporean society. (references) | ||
However, it does not include furniture, fixtures, or other equipment which are not an integral part of the building. (references) | ||
Children | Cuba | State organizations and schools are charged with the integral formation of children and youth. (references) |
Civil Liberties | Ireland | Although religious instruction is an integral part of the curriculum, parents may exempt their children from such instruction. (references) |
Moldova | Travel between Transnistria and the rest of the country was not prevented, and the Government maintained that Transnistria is an integral part of a single state, although with a status yet to be determined. (references) | |
Economic History | Ireland | From 1800 to 1921, Ireland was an integral part of the United Kingdom. (references) |
Malaysia | Offsets and/or technology transfer are usually an integral part of any large deal. (references) | |
Ireland | Process control instruments comprise an integral element of Irish manufacturing processes. (references) | |
Minorities | Turkmenistan | Although most citizens do not emphasize mosque attendance or observance of many Islamic customs practiced in other parts of the Muslim world, they view being Muslim as an integral part of the national culture and of Turkmen identity. (references) |
Political Economy | NETHERLANDS | The government successfully reduced its role in the economy during the 1990s, and structural and regulatory reforms have been an integral component of Dutch economic policy since the early 1980s. (references) |
Finland | This interest is reflected in the U.S. Northern European Initiative (NEI) and the Finnish-inspired EU Northern Dimension program, which was an integral part of the Finnish EU Presidency platform from July through December 1999. (references) | |
Political Rights | Swaziland | Chiefs are an integral part of society and act as overseers or guardians of families within the communities and traditionally report directly to the king. (references) |
Trade | Azerbaijan | Provisions for the importation of goods and equipment are an integral part of PSAs. (references) |
Travel | Albania | Time spent drinking coffee is considered an integral part of the meeting and should not be dismissed as a waste of time. (references) |
Worker Rights | Belarus | The FTUB reported that union members at other Integral plants similarly had been threatened. (references) |
Costa Rica | Nonetheless, child labor remains an integral part of the informal economy, particularly in small-scale agriculture and family-run microenterprises selling various items, which employ a significant proportion of the labor force. (references) | |
Belarus | In September 2000, FTUB members reportedly were pressured by the management of Dzerzhinsky, a subsidiary of the state-owned electronics manufacturer Integral, to break with their union and join a management-established and -run union. (references) | |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| Speaker | Term | Phrase(s) |
Jimmy Carter | 1977-1981 | The human rights policy of the United States has been an integral part of our overall foreign policy for the past several years. |
Source: compiled by the editor from various references. | ||
| "Integral" is generally used as an adjective (general or positive) -- approximately 91.16% of the time. "Integral" is used about 938 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Adjective (general or positive) | 91.16% | 855 | 8,259 |
| Noun (singular) | 8.2% | 77 | 37,929 |
| Noun (proper) | 0.64% | 6 | 143,867 |
| Total | 100.00% | 938 | N/A |
Source: compiled by the editor from several corpora; see credits.
| Country | Name | Country | Name |
| Spain | Compania de Distribucion Integral Logistica S.A. | USA | Integral Systems, Inc. |
| (more examples...) |
Source: compiled by the editor from Icon Group International, Inc.
Expressions using "integral": definite integral ♦ Elliptic integral ♦ Fourier integral ♦ Fourier integral equation ♦ indefinite integral ♦ integral absorbed dose ♦ integral boundary ♦ integral calculus ♦ integral herd ♦ integral key ♦ integral rotor ♦ integral tank ♦ Integral unit ♦ multiple integral. Additional references. | |
| Hyphenated Usage | |
Beginning with "integral": integral-order, integral-with-time-of-absolute-error. | |
Ending with "integral": non-integral. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Language | Translations for "integral"; alternative meanings/domain in parentheses. | |
Afrikaans | volledig (entire, overall, whole), vol (complete, completely, full, fully), totaal (overall, total), hele (entire, overall, whole). (various references) | |
Albanian | integral, i tërë (all, clear, downright, entire, gross, thorough, total, undivided, whole), i plotë (all out, big, broad, clear, close, complete, comprehensive, crowded, dead, direct, dyed in the wool, entire, flush, fraught, full, gross, out and out, outright, overall, perfect, plenary, Plumb, plummy, pregnant, radical, round, sheer, solid, thorough, thoroughgoing, total, unabbreviated, uncut, unquestioning, utter, whole), tërësi (altogether, complex, ensemble, entirety, integer, integrality, integrity, massiveness, plenitude), përbërës (component, composing, compound, constituent, fixings, ingredient, integrate, multiple, multiplex, principle), numër i plotë (integer). (various references) | |
Arabic | كامل (absolute, all out, complete, completed, concluded, consummate, entire, essential, finalized, finished, full, full blown, full scale, integrate, out and out, outright, perfect, plenary, plumb, rounded off, stark, thorough, thoroughgoing, total, unqualified, utter, whole), متكامل (complete, total, whole), متتام (complementary), تكاملي, تام (blank, complete, crass, entire, every, flat, gone, implicit, outright, perfect, perfected, performed, plenary, pure, rank, regular, round, sound, thorough, thoroughgoing, unconditional, unequivocal, unqualified, utter, whole), صحيح غير كسري, التكامل (integration). (various references) | |
Bulgarian | съществен (essential, formal, fundamental, integrant, intimate, intrinsic, material, meaty, radical, substantial, substantive, underlying, vital), цялостен (across the board, all around, all out, complete, intact, integrate, mass, organic, overall, radical, teetotal, thorough, total, whole), цял (cool, entire, full, good, intact, integrate, livelong, overall, perfect, regular, right, round, sheer, solid, thorough, thoroughgoing, thorough-paced, total, unabridged, unbroken, undivided, whole), неразделен (inseparable, integrant, undivided, unshared), неделим (impartible, indivisible, infrangible, unitary, whole), монолитен (monolithic, solid), пълен (absolute, alive, all out, ample, beefy, clear, compendious, complete, corpulent, dead, entire, exhaustive, explicit, fat, fleshy, fraught, full, grand, implicit, intact, integrate, lousy, murky, out and out, outright, overall, overblown, perfect, plenary, portly, profound, pursy, radical, rank, replenished, replete, riddle, right, round, sheer, stark, stout, substantial, teetotal, thoroughgoing, thorough-paced, total, unabbreviated, universal, unmitigated, unqualified, unreserved, utter, vast, very, well fed, whole, whole-footed, whole-hog), интегрален, интеграл (fluent). (various references) | |
Catalan | sencer (entire, overall, whole). (various references) | |
Chinese | 缺一不可 (integrally). (various references) | |
Czech | integrální, nedílný. (various references) | |
Danish | integrale, hel (entire, overall, whole), fuldstændig (absolute, complete), fuld (complete, drunk, full, intoxicated). (various references) | |
Dutch | integraal, ongeschonden, onaangetast. (various references) | |
Esperanto | integra. (various references) | |
Faeroese | reiðiligur, heilur, fullur (complete, full), fullfíggjaður (absolute, complete, full), allur (all the, each, entire, every, every one, everybody, overall, whole). (various references) | |
Farsi | کامل (Absolute, Complete, Culminant, Exact, Intact, Large, Main, Mature, Perfect, Plenary, Stark, Thorough, Unabridged, Unmitigated, Unqualified, Whole), تمام (All, Complete, Entire, Full, Main, Off, Outandout, Thorough, Through, Whole), صحیح (Authentic, Correct, Exact, Good, Right, Safe, True, Valid), انتگرال , درست (Accurate, Authentic, Conscionable, Correct, Entire, Even, Exact, Genuine, Just, Legitimate, Orthodox, Outandout, Perfect, Plumb, Right, Smackdab, Sock, Sound, Straight, Straightforward, True, Trustworthy, Upright, Valid, Veracious, Whole), بی کسر. (various references) | |
Finnish | integraali, täysi (complete, full, perfect, whole), täydellinen (absolute, complete, entire, full, perfect), koko (dimensions, entire, fairly, overall, rather, size, stature, the whole, volume, whole). (various references) | |
French | intégrale, complet. (various references) | |
Frisian | totaal (absolute, complete, overall, total), hielendal (entire, overall, whole), hiel (absolute, complete, entire, overall, total, whole), gânsk (entire, overall, whole), gâns (entire, overall, quite, very, very much, whole), finaal (entire, overall, whole), alles omfiemjend (absolute, complete, overall, total). (various references) | |
German | integral (definite). (various references) | |
Greek | ακέραιοσ (undivided, whole), ακέραιος (intact, whole), ανυπόσπαστος, αναπόσπαστοσ, ολόκληροσ (entire, livelong, total, whole), ολοκληρωτικόσ (thorough going, totalitarian), ολοκλήρωμα. (various references) | |
Hebrew | שלם (absolute, complete, entire, intact, paymaster, perfect, round, scot free, self contained, strict, total, whole), אי ט'רלי, אי ט'רל, אחי" (even, homogeneous, unified, uniform), בלתי פר". (various references) | |
Hungarian | integrál (integrate, intern, to integrate), egész (all, clear, complete, entire, fairly, have the dithers, I just told him, integrate, it doesn't matter a bit, it doesn't matter in the least, livelong, overall, plumb, some, the lot, total, whole), ép (entire, intact, scot free, sound, unhurt, unscathed, whole). (various references) | |
Icelandic | heill (entire, overall, whole). (various references) | |
Indonesian | integral. (various references) | |
Italian | intero (absolute, all, complete, entire, exclusive, full, intact, livelong, overall, round, thorough, total, whole, wide), integrante (integrant), integrale (complete, comprehensive, outright, unabridged, uncut, undivided, wholemeal), tutto (all, all the, altogether, any, anything, each, each one, entire, every, every one, everybody, everything, overall, whole, whole shoot), totale (absolute, all, all out, blank, complete, entire, gross, overall, sheer, sound, summation, total, utter, whole), pieno (bagful, big, brimful, complete, fill, filled, fraught, full, full cargo, full load, full up, instinct, replete, solid, tight), completo (absolute, all, all out, all round, arrant, blank, clear, complete, dead, entire, full, full up, implicit, inclusive, intimate, out and out, outfit, outright, overall, perfect, round, set, sound, stark, suit, thorough, total, totally, unabridged, utter, whole). (various references) | |
Japanese Kanji | 積分 . (various references) | |
Japanese Katakana | せきぶ". (various references) | |
Korean | 완 한 (Complete, Consummate, CPL, Perfect, utter). (various references) | |
Malay | lengkap (absolute, complete). (various references) | |
Manx | co-hym. (various references) | |
Norwegian | integrerende, integral, helhetlig. (various references) | |
Papiamen | yen (complete, full), tur (all the, each, entire, every, every one, everybody, overall, whole), total (overall, total), kompleto (absolute, complete), henter (entire, overall, whole), henté (entire, overall, whole), hèntèr (entire, overall, whole), enter (entire, overall, whole), enté (entire, overall, whole), emter (entire, overall, whole). (various references) | |
Pig Latin | integralay.(various references) | |
Polish | pełny (complete, full), pełen (complete, full), całkowity (entire, overall, whole). (various references) | |
Portuguese | integral (integrate, unabridged, whole). (various references) | |
Romanian | integrant (integrant), integralã, integral (integrate, whole), tot (all, altogether, approximately, entire, equally, everything, integer, likewise, livelong, long, overall, whole), plin (complete, contents, entire, full, high, orotund, paved, replete, repletion, round, solid), compus (composite, compound, integrate, multiple), întreg (clear, complete, entire, exclusive, full, integer, round, safe, sound, total, unabridged, unbroken, unimpaired, unit, whole). (various references) | |
Russian | цельный (all in one, all-in-one, one-piece, unadulterated), целый (all, clear, entire, even, intact, livelong, unbroken, undivided, whole), целочисленный, целое число (integer, whole number), встроенный (a ~ built, a built, built-in, predefined), неотъемлемый (imprescriptible, inalienable, indefeasible), интегральный, интеграл целый, интеграл. (various references) | |
Scottish | l n (complete, filled, full, fulness, governs the g. of n., repletion, satisfied). (various references) | |
Serbo-Croatian | integralni, integralan, integral, sastavni (component, compositional, constituent, contextual, copulative, structural, syndetic). (various references) | |
Spanish | integral (integrate, uncut), entero (all in one, clear, complete, entire, intact, integer, livelong, not castrated, outright, overall, payment, point, resolute, sound, strong, total, unbroken, upright, whole). (various references) | |
Sranan | furu (complete, fill, fill in, fill up, full), eri (entire, overall, whole). (various references) | |
Swahili | -zima (entire, overall, whole), kamili (absolute, complete, exactly). (various references) | |
Swedish | väsentlig (essential, fundamental, intrinsic, pertinent), integral, hel (clear, entire, full, intact, long, overall, total, unbroken, undivided, universal, whole). (various references) | |
Tagalog | puno (boss, chief, complete, full, leader, tree), punô (complete, full). (various references) | |
Thai | ซึ่งเป็นส่วนประกอบสำคัญ, ครบถ้วน. (various references) | |
Turkish | integral, tek parça (one piece, solid, unipartite), tamsayılardan oluşan, tamamlayıcı (adjunct, collateral, complement, complemental, complementary, component, expletive, follow up, integrant, modifier, processor, supplement, supplemental, supplementary), tam şey (integer), tam (absolute, accomplished, according to cocker, accurate, all out, at the time, bang, bang on, blank, clear, complete, consummate, correct, dead, desperately, downright, due, engrained, entire, even, exact, exactly, factual, full, full complement, fully, holo-, implicit, ingrained, intact, intimate, just, literal, mathematical, on time, out and out, outright, overall, perfect, plenary, Plumb, plunk, precise, precisely, prize, prompt, proper, punctual, rank, right, rightdown, root and branch, round, sharp, sheer, simple, slap bang, slick, solid, spot-on, Square, stark, straight, strict, the very, thorough, thoroughgoing, to a t, true, trueborn, unalloyed, unambiguous, unmitigated, unredeemed, unreserved, utter, very, whole), türevi bilinen fonksiyon, tümlev, bütünleyici (integrant, supplemental, supplementary), bütünü oluşturan, bütün şey, bütün (aggregate, all, all out, all over the, altogether, at all, clear, complement, complete, continuum, entire, entirely, every, everything, gross, holo-, omni-, one and only, out and out, overall, pan-, quite, round, sheer, solid, the total, the whole, total, totality, unbroken, undivided, utter, whole, wholly). (various references) | |
Turkmen | integral (r). (various references) | |
Ukrainian | інтегральний, інтеграл, невід'"мний (inalienable, indefeasible, inherent, integrant, necessary), повний (absolute, all out, big, broad, chock-a-block, complete, crass, definitive, entire, fatty, flush, full, orbicular, out and out, outright, overall, overblown, perfect, plenary, profound, pudgy, pure, rotund, round, total, utter, whole). (various references) | |
Vietnamese | cần cho tính to n bộ, cần cho tính nguyên to n bộ. (various references) | |
Yucatec | chuup (complete, full). (various references) | |
| Source: compiled by the editor from various translation references. | ||
| Language | Period | Translations |
| Latin | 500 BCE-Modern | integer. (various references) |
| Medieval Latin | 700-1500 | integralis. (various references) |
| Source: compiled by the editor from various references. | ||
Derivations | |
Words beginning with "integral": integralities, integrality, integrally, integrals. (additional references) | |
Words ending with "integral": nonintegral. (additional references) | |
| |
"Integral" is suggested in spellcheckers for the following: Andegari, Inderal, Ingelram, integal, integr, integra, integrae, integran, integrel, intenral, Intera, Intercal, intergal, intergral, interial, intirgal, intregal. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "integral" (pronounced i"ntugrul or i"nugrul) |
| 3 | -r u l | amoral, ancestral, antiviral, apparel, astral, aural, auroral, austral, Balmoral, barrel, Beryl, boral, Carle, Carol, carrel, cathedral, central, cerebral, choral, coral, feral, floral, gambrel, goral, immoral, imperil, Sorel, Sorrel, spiral, sterile, tetrahedral, intramural, laurel, Loral, minstrel, mistral, mitral, moral, mural, neural, neutral, nostril, octahedral, oral, orchestral, peril, plural, quarrel, rural, scoundrel, several, ventral, vertebral, viral, virile. |
| 3 | -r u l | amoral, ancestral, antiviral, apparel, astral, aural, auroral, austral, Balmoral, barrel, Beryl, boral, Carle, Carol, carrel, cathedral, central, cerebral, choral, coral, feral, floral, gambrel, goral, immoral, imperil, Sorel, Sorrel, spiral, sterile, tetrahedral, intramural, laurel, Loral, minstrel, mistral, mitral, moral, mural, neural, neutral, nostril, octahedral, oral, orchestral, peril, plural, quarrel, rural, scoundrel, several, ventral, vertebral, viral, virile. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
Direct Anagrams: alerting, altering, relating, tanglier, triangle. | |
| Words within the letters "a-e-g-i-l-n-r-t" | |
-1 letter: aligner, atingle, elating, engrail, gelatin, genital, granite, gratine, ingrate, latrine, nargile, ratline, realign, reginal, reliant, retinal, ringlet, tangier, tangler, tearing, tingler, trenail. | |
-2 letters: aiglet, aigret, aliner, angler, antler, argent, earing, eating, engirt, entail, gainer, gaiter, garnet, gelant, gelati, genial, gentil, glaire, gratin, ingate, larine, learnt, ligate, linage, linear, linger, linter, nailer, ratine. | |
| Words containing the letters "a-e-g-i-l-n-r-t" | |
+1 letter: antiglare, earthling, faltering, gnarliest, granulite, haltering, integrals, lathering, paltering, reflating, replating, reslating, retailing, trameling, traveling, treadling, triangles. | |
+2 letters: blathering, blattering, clattering, corelating, earthlings, flattering, generalist, generality, granulites, integrable, integrally, lacerating, lateraling, laterizing, laureating, leathering, liberating, martingale, multirange, plastering, pregenital, realtering, refloating, regelating, regimental, regulating, regulation, relegating, relegation, relocating, replanting, resaluting, retackling, retailings, slathering, starveling, tolerating, trailering, tramelling, trammeling, travelling, trigeminal, ulcerating, urogenital. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Synonyms 3. Crosswords 4. Usage: Commercial | 5. Images: Slideshow 6. Images: Photo Album 7. Quotations: Historic 8. Quotations: Fiction | 9. Quotations: Non-fiction 10. Quotations: Speeches 11. Usage Frequency 12. Names: Company Usage | 13. Expressions 14. Expressions: Internet 15. Translations: Modern 16. Translations: Ancient | 17. Abbreviations 18. Acronyms 19. Derivations 20. Rhymes | 21. Anagrams 22. Bibliography |
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