intégrer [ɛ͂tegʀe] VERB trans, intr. Verbtabelle anzeigen. intégrer UNIV ugs. 3. würdigt Alan Johnstons bisherige Arbeit als überaus integrer Journalist, der seit 16 Jahren für die BBC tätig ist, vor allem seine Arbeit in den letzten drei. crazylady.eu | Übersetzungen für 'integrer' im Englisch-Deutsch-Wörterbuch, mit echten Sprachaufnahmen, Illustrationen, Beugungsformen.
integrer (Deutsch). Wortart: Komparativ. Silbentrennung: in|te|grer. Aussprache/Betonung: IPA: [ɪnˈteːɡʀɐ]. Grammatische Merkmale: Komparativ von integer. Prädikative und adverbielle Form des Komparativs des Adjektivs integer. integrer ist eine flektierte Form von integer. Die gesamte Deklination findest du auf der. Bedeutung. Info. unbescholten, moralisch einwandfrei; unbestechlich. Beispiele. ein integrer Mann, Politiker; er hatte bis dahin als absolut integer gegolten. Wörterbuch. Integration. Substantiv, feminin – 1. Einbeziehung, Eingliederung in ein größeres 2. [Wieder]herstellung einer Einheit [aus Differenziertem]; 3. intégrer [ɛ͂tegʀe] VERB trans, intr. Verbtabelle anzeigen. intégrer UNIV ugs. Übersetzungen für INTEGRER im Französisch» Deutsch-Wörterbuch von PONS Online:intégrer, intégrer un livre à dans une liste, un ordinateur avec carte son. 3. würdigt Alan Johnstons bisherige Arbeit als überaus integrer Journalist, der seit 16 Jahren für die BBC tätig ist, vor allem seine Arbeit in den letzten drei.
intégrer [ɛ͂tegʀe] VERB trans, intr. Verbtabelle anzeigen. intégrer UNIV ugs. crazylady.eu | Übersetzungen für 'integrer' im Englisch-Deutsch-Wörterbuch, mit echten Sprachaufnahmen, Illustrationen, Beugungsformen. Prädikative und adverbielle Form des Komparativs des Adjektivs integer. integrer ist eine flektierte Form von integer. Die gesamte Deklination findest du auf der.
Integrer - RechtschreibungEs ist ein Brauch von alters her: Wer Sorgen hat, hat auch Likör! Slowenisch Wörterbücher. Konrad Duden.
Using more steps produces a closer approximation, but will always be too high and will never be exact. Alternatively, replacing these subintervals by ones with the left end height of each piece, we will get an approximation that is too low: for example, with twelve such subintervals, the approximate value for the area is 0.
The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely many fine, or infinitesimal steps.
Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit of weighted sums, so that the dx suggested the limit of a difference namely, the interval width.
Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral , which is founded on an ability to extend the idea of "measure" in much more flexible ways.
Thus the notation. Here A denotes the region of integration. There are many ways of formally defining an integral, not all of which are equivalent.
The differences exist mostly to deal with differing special cases which may not be possible to integrate under other definitions, but also occasionally for pedagogical reasons.
The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals. The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval.
A Riemann sum of a function f with respect to such a tagged partition is defined as. The Riemann integral of a function f over the interval [ a , b ] is equal to S if:.
When the chosen tags give the maximum respectively, minimum value of each interval, the Riemann sum becomes an upper respectively, lower Darboux sum , suggesting the close connection between the Riemann integral and the Darboux integral.
It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem.
Then the integral of the solution function should be the limit of the integrals of the approximations.
However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral.
Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated. Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same.
Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel :.
I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum.
This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor.
This is my integral. As Folland puts it, "To compute the Riemann integral of f , one partitions the domain [ a , b ] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f ".
In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.
The Lebesgue integral of f is then defined by. A general measurable function f is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of f and the x -axis is finite:.
In that case, the integral is, as in the Riemannian case, the difference between the area above the x -axis and the area below the x -axis:.
The collection of Riemann-integrable functions on a closed interval [ a , b ] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration.
Thus, firstly, the collection of integrable functions is closed under taking linear combinations ; and, secondly, the integral of a linear combination is the linear combination of the integrals:.
In this situation, the linearity holds for the subspace of functions whose integral is an element of V i. Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral.
This is the approach of Daniell for the case of real-valued functions on a set X , generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space.
See Hildebrandt for an axiomatic characterization of the integral. A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [ a , b ] and can be generalized to other notions of integral Lebesgue and Daniell.
In this section, f is a real-valued Riemann-integrable function. The integral. The values a and b , the end-points of the interval , are called the limits of integration of f.
The first convention is necessary in consideration of taking integrals over subintervals of [ a , b ] ; the second says that an integral taken over a degenerate interval, or a point , should be zero.
One reason for the first convention is that the integrability of f on an interval [ a , b ] implies that f is integrable on any subinterval [ c , d ] , but in particular integrals have the property that:.
The fundamental theorem of calculus is the statement that differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved.
An important consequence, sometimes called the second fundamental theorem of calculus , allows one to compute integrals by using an antiderivative of the function to be integrated.
Let f be a continuous real-valued function defined on a closed interval [ a , b ]. Let F be the function defined, for all x in [ a , b ] , by.
Then, F is continuous on [ a , b ] , differentiable on the open interval a , b , and. Let f be a real-valued function defined on a closed interval [ a , b ] that admits an antiderivative F on [ a , b ].
That is, f and F are functions such that for all x in [ a , b ] ,. If f is integrable on [ a , b ] then. The second fundamental theorem allows many integrals to be calculated explicitly.
For example, to calculate the integral. Then the value of the integral in question is. Tables of this and similar antiderivatives can be used to calculate integrals explicitly, in much the same way that derivatives may be obtained from tables.
Integrals are used extensively in many areas. For example, in probability theory , they are used to determine the probability of some random variable falling within a certain range.
Integrals can be used for computing the area of a two-dimensional region that has a curved boundary, as well as computing the volume of a three-dimensional object that has a curved boundary.
The area of a two-dimensional region can be calculated using the aforementioned definite integral. Integrals are also used in physics, in areas like kinematics to find quantities like displacement , time , and velocity.
Integrals are also used in thermodynamics , where thermodynamic integration is used to calculate the difference in free energy between two given states.
A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration.
An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals.
If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity:.
If the integrand is only defined or finite on a half-open interval, for instance a , b ] , then again a limit may provide a finite result:.
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x -axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane that contains its domain.
This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over R uses a double integral sign:.
Integration over more general domains is possible. The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces.
Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields.
A line integral sometimes called a path integral is an integral where the function to be integrated is evaluated along a curve. In the case of a closed curve it is also called a contour integral.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve.
This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force , F , multiplied by displacement, s , may be expressed in terms of vector quantities as:.
This gives the line integral. A surface integral generalizes double integrals to integration over a surface which may be a curved set in space ; it can be thought of as the double integral analog of the line integral.
The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.
For an example of applications of surface integrals, consider a vector field v on a surface S ; that is, for each point x in S , v x is a vector.
Imagine that we have a fluid flowing through S , such that v x determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time.
To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, which we integrate over the surface:.
The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux.
Thus surface integrals have applications in physics, particularly with the classical theory of electromagnetism. In complex analysis , the integrand is a complex-valued function of a complex variable z instead of a real function of a real variable x.
This is known as a contour integral. A differential form is a mathematical concept in the fields of multivariable calculus , differential topology , and tensors.
Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as:. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral.
Here the basic differentials dx , dy , dz measure infinitesimal oriented lengths parallel to the three coordinate axes. Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds curves, surfaces, and their higher-dimensional analogs.
The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem , Green's theorem , and the Kelvin-Stokes theorem.
The discrete equivalent of integration is summation. Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time scale calculus.
The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus.
Let f x be the function of x to be integrated over a given interval [ a , b ]. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus,.
The integral is not actually the antiderivative, but the fundamental theorem provides a way to use antiderivatives to evaluate definite integrals.
The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative.
More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals.
Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include: Basic methods - These are the fundamental methods and are necessary to know to integrate any function.
Derived methods - These are methods derived from the basic methods to make the process of integration easier for some functions.
Alternative methods exist to compute more complex integrals. Many nonelementary integrals can be expanded in a Taylor series and integrated term by term.
Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-functions can also be used, assuming that the integrand can be written as a product of Meijer G-functions.
There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum.
Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral. Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.
Specific results which have been worked out by various techniques are collected in the list of integrals. Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired.
Extensive tables of integrals have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to computer algebra systems that are specifically designed to perform difficult or tedious tasks, including integration.
Symbolic integration has been one of the motivations for the development of the first such systems, like Macsyma and Maple.
A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist.
The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it.
Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule.
Consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function.
On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists.
The Risch algorithm, implemented in Mathematica , Maple and other computer algebra systems , does just that for functions and antiderivatives built from rational functions, radicals , logarithm, and exponential functions.
Some special integrands occur often enough to warrant special study. Extending the Risch's algorithm to include such functions is possible but challenging and has been an active research subject.
More recently a new approach has emerged, using D -finite functions , which are the solutions of linear differential equations with polynomial coefficients.
Most of the elementary and special functions are D -finite, and the integral of a D -finite function is also a D -finite function.
This provides an algorithm to express the antiderivative of a D -finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a D -function as the sum of a series given by the first coefficients, and provides an algorithm to compute any coefficient.
Some integrals found in real applications can be computed by closed-form antiderivatives. Others are not so accommodating. Some antiderivatives do not have closed forms, some closed forms require special functions that themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow.
This motivates the study and application of numerical approximations of integrals. This subject, called numerical integration or numerical quadrature , arose early in the study of integration for the purpose of making hand calculations.
The development of general-purpose computers made numerical integration more practical and drove a desire for improvements.
The goals of numerical integration are accuracy, reliability, efficiency, and generality, and sophisticated modern methods can vastly outperform a naive method by all four measures.
Using the left end of each piece, the rectangle method sums 16 function values and multiplies by the step width, h , here 0. The accuracy is not impressive, but calculus formally uses pieces of infinitesimal width, so initially this may seem little cause for concern.
Indeed, repeatedly doubling the number of steps eventually produces an approximation of 3. However, 2 18 pieces are required, a great computational expense for such little accuracy; and a reach for greater accuracy can force steps so small that arithmetic precision becomes an obstacle.
A better approach replaces the rectangles used in a Riemann sum with trapezoids. The trapezoid rule sums all 17 function values, but weights the first and last by one half, and again multiplies by the step width.
This immediately improves the approximation to 3. Furthermore, only 2 10 pieces are needed to achieve 3. The idea behind the trapezoid rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further.
Simpson's rule approximates the integrand by a piecewise quadratic function. Riemann sums, the trapezoid rule, and Simpson's rule are examples of a family of quadrature rules called Newton—Cotes formulas.
The degree n Newton—Cotes quadrature rule approximates the polynomial on each subinterval by a degree n polynomial.
This polynomial is chosen to interpolate the values of the function on the interval. Commutative algebra.
Noncommutative algebra. Main article: Integer computer science. Mathematics portal. In Bach, Emmon W. Quantification in Natural Languages.
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Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve. Look up integer in Wiktionary, the free dictionary.crazylady.eu | Übersetzungen für 'integrer' im Englisch-Deutsch-Wörterbuch, mit echten Sprachaufnahmen, Illustrationen, Beugungsformen. Ein integrer Mensch lebt und handelt in dem Bewusstsein, dass sich seine persönlichen Überzeugungen, Maßstäbe und Wertvorstellungen in seinem Verhalten. Lernen Sie die Übersetzung für 'integrer' in LEOs Englisch ⇔ Deutsch Wörterbuch. Mit Flexionstabellen der verschiedenen Fälle und Zeiten ✓ Aussprache und. Rechtschreibung gestern und heute. Haar, Faden und Damoklesschwert. Zahlen und Ziffern. Griechisch Wörterbücher. Worttrennung in te Endzeitfilme Beispiel ein in Paterson Film g rer Charakter. Leichte-Sprache-Preis
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If the natural numbers are identified with the corresponding integers using the embedding mentioned above , this convention creates no ambiguity.
In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines.
Integers are represented as algebraic terms built using a few basic operations e. There exist at least ten such constructions of signed integers.
This operation is not free since the integer 0 can be written pair 0,0 , or pair 1,1 , or pair 2,2 , etc.
This technique of construction is used by the proof assistant Isabelle ; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.
An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity.
It is, however, certainly possible for a computer to determine whether an integer value is truly positive.
Fixed length integer approximation data types or subsets are denoted int or Integer in several programming languages such as Algol68 , C , Java , Delphi , etc.
Variable-length representations of integers, such as bignums , can store any integer that fits in the computer's memory. From Wikipedia, the free encyclopedia.
For computer representation, see Integer computer science. For the generalization in algebraic number theory, see Algebraic integer.
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