integral [maths]

8/8/2019 INTEGRAL [MATHS]

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NAME: TERM PROJECT OF MATHSSECTION-REG-NO-ROLLNO-TERM PROJECT-MATHAMATICS

INTEGRAL

International Gamma-Ray Astrophysics

Laboratory (INTEGRAL)

Artist's illustration of INTEGRAL in orbit (credit: ESA)

General information

Organization ESA / NASA / RKA

Launch date 17 October2002

Mass over 4,000 kg

Orbit height 9,000 km (perigee)

153,000 km (apogee)


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Orbit period 72 hr

Telescope style coded mask

Wavelength gamma ray

Diameter 3.7 m

Collecting area 500 cm (SPI, JEM-X)

3,100 cm (IBIS)

Focal length ~4 meters

Instruments

SPI SpecTROMETER

IBIS Imager

JEM-X X-ray monitor

OMC optical monitor

The EuropEON SPACE AGENCY INTErnational Gamma-Ray AstrophysicsLaboratory (INTEGRAL) is detecting some of the most energetic radiation that comesfrom space. It is the most sensitive gammaRAY observatory ever launched.

INTEGRAL is an ESA mission in cooperation with the Russian and NASA It has hadsome notable successes, for example in detecting a mysterious '. It has also had great

success in investigating and evidence for black holes

Instruments :

Four instruments are coaligned to study a target across several ranges. The coded maskswere led by the University of Valencia, Spain.


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The INTEGRAL imager, IBIS (Imager on-Board the INTEGRAL Satellite) observesfrom 15 keV (hard X-rays) to 10 MeV(gamma rays). Mechanical resolution is 12 arcmin,but deconvolution can reduce that to as little as 1 arcmin. A 95 x 95 mask of rectangulartungsten tiles sits 3.2 meters above the detectors. The detector system contains a forwardplane of 128 x 128 Cadmium-Telluride tiles (ISGRI- Integral Soft Gamma-Ray Imager),

backed by a 64 x 64 plane of Caesium-Iodide tiles (PICsIT- Pixellated Caesium-IodideTelescope). ISGRI is sensitive up to 500 keV, while PICsIT extends to 10 MeV. Both aresurrounded by passive shields of tungsten and lead.

The primary spectrometer aboard INTEGRAL is SPI, the SPectrometer for INTEGRAL.It observes radiation between 20 keV and 8 MeV. SPI consists of a coded mask ofhexagonal tungsten tiles, above a detector plane of 19 germanium crystals (also packedhexagonally). The Ge crystals are actively cooled with a mechanical system, and give anenergy resolution of 2 keV at 1 MeV.

IBIS and SPI need a method to stop background radiation. The SPI ACS

(AntiCoincidence Shield) consists of a mask shield and a detector shield. The mask shieldis a layer of plastic scintillatorbehind the tungsten tiles. It absorbs secondary radiationproduced by impacts on the tungsten. The rest of the shield consists ofBGO scintillatortiles around the sides and back of the SPI.

The enormous area of the ACS that results makes it an instrument in its own right. Its all-sky coverage and sensitivity make it a natural gamma-ray burst detector, and a valuedcomponent of the IPN (InterPlanetary Network). Recently, new algorithms allow theACS to act as a telescope, through double Compton scattering. Thus ACS can studyobjects outside the field of view of the other instruments, with surprising spatial andenergy resolution.

Dual JEM-X units provide additional information on targets. They observe in soft andhard X-rays, from 3 to 35 keV. Aside from broadening the spectral coverage, imaging ismore precise due to the shorter wavelength. Detectors are gas scintillators (xenon plusmethane) in a microstrip layout, below a mask of hexagonal tiles.

INTEGRAL mounts an Optical Monitor (OM), sensitive from 500 to 850nm. It acts asboth a framing aid, and can note the activity and state of some brighter targets.

The spacecraft also mounts a radiation monitor, INTEGRAL Radiation EnvironmentMonitor (IREM), to note the orbital background for calibration purposes. IREM has an

electron and a proton channel, though radiation up to cosmic rayscan be sensed. Shouldthe background exceed a preset threshold, IREM can shut down the instrument.

An integral helps to find out how much space is under a graph of something. Integralsundo derivatives. A derivative helps to find what the steepness is of a graph.

This is the symbol for integration:


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Integrals and derivatives are part of a branch ofmathematics called calculus.

Integration helps when trying to multiply units into a problem. For example, if a problem

with rate ( ) needs an answer with just distance, one solution is to integrate

with respect to time. This means multiplying in time (to cancel the time in ).This is done by adding small slices of the rate graph together. The slices are close to zeroin width, but adding them forever makes them add up to a whole. This is called aRiemann Sum.

Adding these slices together gives the equation that the first equation is the derivative of.Integrals are kind of like adding machines.

Another time integration is helpful is when finding the volume of a solid. It can add two-

dimensional (without width) slices of the solid together forever until there is a width.This means the object now has three dimensions: the original two and a width. This givesthe volume of the three-dimensional object described.

Abelian integral :

In mathematics, an abelian integral in Riemann surface theory is a function related tothe indefinite integral of a differential of the first kind. Suppose we are given a Riemannsurface Sand on it a differential 1-form that is everywhere holomorphic on S, and fix apointPon Sfrom which to integrate. We can regard

as a multi-valued functionf(Q), or (better) an honest function of the chosen path Cdrawnon SfromPto Q. Since Swill in general be multiply-connected, one should specify C,but the value will in fact only depend on the homology class ofC.

In the case ofSa compact Riemann surface ofgenus 1, i.e. an elliptic curve, suchfunctions are the elliptic integrals. Logically speaking, therefore, an abelian integralshould be a function such asf.

Such functions were first introduced to study hyperelliptic integrals, i.e. for the casewhere Sis a hyperelliptic curve. This is a natural step in the theory of integration to thecase of integrals involving algebraic functions A, whereA is a polynomial of degree > 4.The first major insights of the theory were given by Niels Abel; it was later formulated interms of the Jacobian varietyJ(S). Choice ofPgives rise to a standard holomorphicmapping


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SJ(S)

ofcomplex manifolds. It has the defining property that the holomorphic 1-forms onJ(S),of which there aregindependent ones ifgis the genus ofS, pull backto a basis forthe differentials of the first kind on S. Elliptic integral

Redirected from Elliptic Integral

An elliptic integral is any functionfwhich can be expressed in the form f(x) = \int_{c}^{x} R(t,P(t))\ dt

whereR is a rational function of its two arguments,Pis the square root of apolynomialof degree 3 or 4 with no repeated roots, and c is a constant.

Particular examples include:

The complete elliptic integral of the first kindKis defined as K(x) = \int_{0}^{1} \frac{1}{ \sqrt{(1-t^2)(1-x^2 t^2)} }\ dt and can be computed in terms of the arithmetic-geometric mean.

The complete elliptic integral of the second kindEis defined as

E(x) = \int_{0}^{1} \frac{ \sqrt{1-x^2 t^2} }{ \sqrt{1-t^2} }\ dt

A real numberis called integral if it is an integer. The integral value of a real numberxis defined to be the largest integer which is less than or equal tox; it is often denoted byx and also called the floor function.

Integral of a mathematical function :

In the integral calculus, the integral of a function is informally defined as the size of thearea delimited by the x axis and the graph of the function. In the case of non-negativefunctions, the notion of area is the usual one. For functions which take negative values, a

special interpretation is used, and "negative area" is possible.

Letf(x) be a function of the interval [a,b] into the real numbers. For simplicity, assumethat this function is non-negative (it takes no negative values.) The set S=Sf:={(x,y)|


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0yf(x)} is the region of the plane betweenfand thex axis. Measuring the "area" ofSisdesirable, and this area is denoted by f, and it is the (definite) integral off.

Details can be found underRiemann integral and Lebesgue integral. The concept ofRiemann 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 isRiemann integrable, then it is also Lebesgue integrable, and the two integrals coincide.

The antiderivative approach occurs when we seek to find a functionF(x) whosederivativeF(x) is some given function f(x). This approach is motivated by calculus, andis 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 Lebesgueintegrable.) The nonobvious theorem that states that the two approaches ("area under thecurve" and "antiderivative") are in some sense the same as the fundamental theorem of

calculus

The nuance between Riemann and Lebesgue integration :

Both the Riemann and the Lebesgue integral are approaches to integration which seek tomeasure 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 ofmeasuring the area under the curve. In the case of the Riemann integral, this choice is sothat the area under the curve can be regarded as a finite union of rectangles, and thefunctions are calledstep functions. For the Lebesgue integral, "rectangle" is replaced bysomething more sophisticated, and the resulting functions are calledsimple functions.

Then we try to impose monotonicity. If 0fg(and hence Sf is a subset ofSg) then we

should have that fg. With this monotonicity requirement, for an arbitrary non-negativefunctionf, we can approximate its area from below using a carefully chosen elementaryfunctions (in the case of Riemann integration, a step function, and in the case ofLebesgue integration, a simple function.) We chooses so thatsfbuts is very close tof.The area unders is a lower bound for the integral off, and it is called a lower sum. In thecase of the Riemann integral, we also produce upper sums in a similar fashion: we choosestep functions, says, so thatsfbuts is very close tof, and we regard such an upper sumas an upper bound for the area underf. The Lebesgue theory does not use upper sums.


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Lastly, a limit-taking step is taken to make the elementary functions approachfmore andmore closely, and an area is obtained for some functionsf. The functions which we canintegrate are said to be integrable. However, the differences begin here; the Riemanntheory was simpler thus far, but its simplicity results in a more limited set of integrablefunctions than the Lebesgue theory. In addition, the interaction between limits and the

integral are more difficult to describe in the Riemann setting.

In branch of mathematics known as Real analysis, the

Riemann integral is a simple way of viewing the integral of

a function on an interval as the area under the curve.

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

Letf(x) be a real-valued function of the interval [a,b], so that for allx,f(x)0 (fis non-negative.) Further let S=Sf:={(x,y)|0yf(x)} (see Figure 2) be the region of the planeunder the functionf(x) and above the interval [a,b]. We are interested in measuring thearea ofSif that is possible, and we denote this area by abf(x)dx. In case several variablesappear inf, the dx will serve to specify the variable of integration. If the variable ofintegration and interval of integration are understood, the notation can be simplified to f.

Once we have succeeded in evaluating the integral offfor certainfwhich are non-negative, we can extend the integral to functions which may take negative values bylinearity. Some functions have no clear Riemann integral, but especially, the interactionsoflimits and the Riemann integral are difficult to study. An improvement is to use theLebesgue integral which both succeeds at integrating a broader variety of functions, aswell as better describing the interactions oflimits and integrals.

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

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


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

For certain functions, the theory ofantiderivatives provides exact results for definiteintegrals. While the Riemann integration theory justifies taking limits and provides ageometric point of view, the antiderivative theory of integration gives tools forintegrating certain formulae precisely.

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

Step functions :Let E be any subset of [a,b]. LetXE(x) be the function which is 1 ifx is in E and 0 ifx isnot in E.XE is called the indicating function of E, or the characteristic function of E.

These functions are our starting point, and we should agree that

zX[c,d](x)dx=z(d-c)

for any interval [c,d] in [a,b] and any constantz0. Indeed, in this case, the area under thecurve is a rectangle with base [c,d] and heightz.

Likewise, some geometric experimentation with such functions suggests that iff1,f2, ...,fnare n indicating functions of intervals, a_1, a_2, ..., a_n are scalars, thenthe area under

f=a_1f_1+a_2f_2+...+a_nf_n

should be

\int f = a_1 \int f_1 + a_2 \int f_2 + ... + a_n \int f_n

A function of this form is called a linear combination of indicating functions, or moresimply astep function. We note now that we've decided what the integral ofstepfunctions ought to be.

We will take a shortcut now by stating that the preceding formula will be used even ifsome (or all) of the coefficients aj are negative.


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A crucial difference between the Riemann integral and the Lebesgue integral is that thestep functions of the Lebesgue integral are linear combinations of indicating functions ofsets which are not necessarely intervals. Of course, work is then required to calculate theintegral of this larger class of step functions. Also note that the Lebesgue integral doesnot use upper sums, and that non-negative functions are dealt with first, before extending

to functions which may take negative values.

Lower and upper sums :

From the geometry of the problem, we impose that iff(x)g(x) for allx in [a,b] then wereally ought to have

fgsimply by seeing that the area Sf is a subset of Sg (at least in the case of non-negativefunctions, this is clear.) We call this requirement monotonicity.

Given the integral ofstep functions and the monotonicity requirement, we can get a firststab at integrating arbitrary non-negative functions. Letf(x) be a real-valued function of[a,b] and let l(x) be a step function such that l(x)f(x) for allx. Furthermore, let u(x) be astep function such that u(x)f(x) for allx. If we are to assign a value to f consistent withthe monotonicity requirement, then we need that

lfuThe integral lis then called a lower sum forfand the integral u is then called an uppersum forf. The preceding inequality must hold for all lower and upper sums off, so wecan deduce another inequality:

supllfinfuu

where supllis the smallest upper bound for all lower sums, and infuu is the largest lowerbound for all upper sums (see supremum and infimum.) The number supllis sometimescalled the lower sum; likewise, the number infuu is the upper sum.

If the supremum and infimum are equal, then there is only one choice left for f. It maynot happen that the supremum is larger than the infimum (this is by our construction, asthe reader may check.) However, it may happen that the supremum is less, and not equalto, the infimum. For instance, the reader may check that, for the indicating function

XQ


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where Q is the set ofrational numbers in [a,b], a0.

The collection of functions whose lower sum and upper sum are equal and finite is the setof Riemann integrable functions. By contrast, functions that have differing upper and

lower sums are said to be non-Riemann integrable. In the context of this article, we willsay integrable or non-integrable with the understanding that we are speaking of Riemannintegrability.

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

Results about the Riemann integral :

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

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

Theorem 2: Any real-valued continuous function of the interval [a,b] is integrable.Theproof relies on the fact that any continuous function of an interval is necessarelyuniformly continuous.

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

The boundedness requirement can not be dropped.

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

Corollary 5: Let C(a,b) be the Banach space of continuous functions over[a,b] with theuniform norm[?]. Then I:ff is continuous. Together with Lemma 1, this says that theintegral is a continuous functional of C(a,b).

The hypotheses of theorem 4 (uniform convergence on a fixed, bounded interval) arevery strong. A primary failing of the Riemann integral is the difficulty we face whenattempting to relax these hypotheses. In fact, the numericalsequence fk will converge tothe number fa lot more often than is suggested by the theorem, but it is very difficult toprove so in this setting. The correct way of getting a stronger theorem is to use theLebesgue integral.
http://opt/scribd/conversion/tmp/scratch10096/Uniform%20normhttp://opt/scribd/conversion/tmp/scratch10096/Uniform%20norm


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Another problem with the Riemann integral is that it does not extend to unboundedintervals very succesfully. If we wish to integrate a functionffrom - to +, we cannaively calculate

limn+ -nnf(x)dx

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

INTEGRAL :In the branch ofmathematics known as real analysis, the Riemann integral, created byBernhard Riemann, was the first rigorous definition of the integral of a function on aninterval. While the Riemann integral is unsuitable for many theoretical purposes, it is oneof the easiest integrals to define. Some of these technical deficiencies can be remedied bythe Riemann-Stieltjes integral, and most of them disappear in the Lebesgue integral.

The integral as the area of a region under a curve.

INTRODUCTION :

Letf(x) be a non-negative real-valued function of the interval [a,b], and let S= {(x,y) | 0


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The basic idea of the Riemann integral is to use very simple approximations for the areaofS. By taking better and better approximations, we can say that "in the limit" we get

exactly the area ofSunder the curve.

A sequence of Riemann sums. The numbers in the upper right are the areas of the greyrectangles. They converge to the integral of the function.

Definition of the Riemann integralPartitions of aninterval:

Apartition of an interval [a,b] is a finite sequence. Each [xi,xi + 1] is called asubintervalof the

partition. The mesh of a partition is defined to be the length of the longest subinterval[xi,xi + 1], that is, it is max(xi + 1 xi) where . It is also called the norm ofthe partition.

A tagged partition of an intervalis a partition of an interval together with a finite

sequence of numbers subject to the conditions that for each i,

. In other words, it is a partition together with a distinguished point ofevery subinterval. The mesh of a tagged partition is defined the same as for an ordinarypartition.

Suppose that together with are a tagged partition of [a,b], andthat together with are another tagged partition of [a,b]. Wesay that and together are a refinementof

together with if for each integeri with , there is an integerr(i)

such thatxi =yr(i) and such that ti =sj for somej with . Said moresimply, a refinement of a tagged partition takes the starting partition and adds more tags,but does not take any away.

We can define a partial orderon the set of all tagged partitions by saying that one taggedpartition is bigger than another if the bigger one is a refinement of the smaller one.


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Riemann sums:

Choose a real-valued functionfwhich is defined on the interval [a,b]. TheRiemann sum

offwith respect to the tagged partition together with is:

Each term in the sum is the product of the value of the function at a given point and thelength of an interval. Consequently, each term represents the area of a rectangle withheightf(ti) and lengthxi + 1 xi. The Riemann sum is the signed area under all therectangles.

The Riemann integral:

Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function asthe partitions get finer and finer. However, being precise about what is meant by "finerand finer" is somewhat tricky.

One important fact is that the mesh of the partitions must become smaller and smaller, sothat in the limit, it is zero. If this were not so, then we would not be getting a goodapproximation to the function on certain subintervals. In fact, this is enough to define anintegral. To be specific, we say that the Riemann integral offequalss if the followingcondition holds:

For all > 0, there exists > 0 such that for any tagged partition andwhose mesh is less than , we have

However, there is an unfortunate problem with this definition: it is very difficult to workwith. So we will make an alternate definition of the Riemann integral which is easier towork with, then prove that it is the same as the definition we have just made. Our newdefinition says that the Riemann integral offequalss if the following condition holds:

For all > 0, there exists a tagged partition and suchthat for any refinement and of and

, we have


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Both of these mean that eventually, the Riemann sum offwith respect to any partitiongets trapped close tos. Since this is true no matter how close we demand the sums betrapped, we say that the Riemann sums converge tos. These definitions are actually aspecial case of a more general concept, a net.

As we stated earlier, these two definitions are equivalent. In other words,s works in thefirst definition if and only ifs works in the second definition. To show that the firstdefinition implies the second, start with an , and choose a that satisfies the condition.Choose any tagged partition whose mesh is less than . Its Riemann sum is within ofs,and any refinement of this partition will also have mesh less than , so the Riemann sumof the refinement will also be within ofs. To show that the second definition implies thefirst, it is easiest to use the Darboux integral. First one shows that the second definition isequivalent to the definition of the Darboux integral; for this see the page on Darbouxintegration. Now we will show that a Darboux integrable function satisfies the firstdefinition. Choose a partition such that the lower and upper Darboux sums

with respect to this partition are within of the values of the Darboux integral. Let requal , where Mi and mi are the supremum and infimum,

respectively, offon [xi,xi + 1], and let be less than both and .Then it is not hard to see that the Riemann sum offwith respect to any tagged partition of

mesh less than will be within of the upper or lower Darboux sum, so it will be within ofs.

Examples:Let be the function which takes the value 1 at every point. AnyRiemann sum offon [0,1] will have the value 1, therefore the Riemann integral offon[0,1] is 1.

Let be the indicator function of the rational numbers in [0,1]; that is,takes the value 1 on rational numbers and 0 on irrational numbers. This function does nothave a Riemann integral. To prove this, we will show how to construct tagged partitionswhose Riemann sums get arbitrarily close to both zero and one.

To start, let and be a tagged partition (each ti is betweenxiandxi + 1). Choose > 0. The ti have already been chosen, and we can't change the valueoffat those points. But if we cut the partition into tiny pieces around each ti, we canminimize the effect of the ti. Then, by carefully choosing the new tags, we can make thevalue of the Riemann sum turn out to be within of either zero or oneour choice!


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Our first step is to cut up the partition. There are n of the ti, and we want their total effectto be less than . If we confine each of them to an interval of length less than / n, thenthe contribution of each ti to the Riemann sum will be at least and at most. This makes thetotal sum at least zero and at most . So let be a positive number less than / n. If ithappens that two of the ti are within of each other, choose smaller. If it happens that

some ti is within of somexj, and ti is not equal toxj, choose smaller. Since there areonly finitely many ti andxj, we can always choose sufficiently small.

Now we add two cuts to the partition for each ti. One of the cuts will be at ti / 2, andthe other will be at ti + / 2. If one of these leaves the interval [0,1], then we leave it out.ti will be the tag corresponding to the subinterval [ti / 2,ti + / 2]. Ifti is directly ontop of one of thexj, then we let ti be the tag for both [ti / 2,xj] and [xj,ti + / 2]. Westill have to choose tags for the other subintervals. We will choose them in two differentways. The first way is to always choose a rational point, so that the Riemann sum is aslarge as possible. This will make the value of the Riemann sum at least 1 . The secondway 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 .Since we started from an arbitrary partition and ended up as close as we wanted to eitherzero or one, it is false to say that we are eventually trapped near some numbers, so thisfunction is not Riemann integrable. However, it is Lebesgue integrable. In the Lebesguesense its integral is zero, since the function is zero almost everywhere. But this is a factthat is beyond the reach of the Riemann integral.

Other concepts similar to the Riemann integral:

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

Some calculus books do not use general tagged partitions, but limit themselves to specifictypes 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 aleft-hand Riemann sum, ti =xi for all i, and in a right-hand Riemann sum, ti =xi + 1 for all

i. Alone this restriction does not impose a problem: we can refine any partition in a waythat makes it a left-hand or right-hand sum by subdividing it at each ti. In more formallanguage, the set of all left-hand Riemann sums and the set of all right-hand Riemannsums is cofinal in the set of all tagged partitions.

Another popular restriction is the use of regular subdivisions of an interval. For example,the n'th regular subdivision of [0,1] consists of the intervals. Again, alone this restriction


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

However, combining these restrictions, so that one uses only left-hand or right-handRiemann 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 theintegral. But under these conditions the indicator function will appear to be integrable on[0,1] with integral equal to one: Every endpoint of every subinterval will be a rationalnumber, so the function will always be evaluated at rational numbers, and hence it willappear to always equal one. The problem with this definition becomes apparent when wetry to split the integral into two pieces. The following equation ought to hold:If we useregular subdivisions and left-hand or right-hand Riemann sums, then the two terms on theleft are equal to zero, since every endpoint except 0 and 1 will be irrational, but as wehave seen the term on the right will equal 1.As defined above, the Riemann integralavoids this problem by refusing to integrate. The Lebesgue integral is defined in such away that all these integrals are 0.

Facts about the Riemann integral:

The Riemann integral is a linear transformation; that is, iffandgare Riemann-integrableon [a,b] and and are constants, then A real-valued functionfon [a,b] is Riemann-integrable if and only if it isbounded and continuous almost everywhere. If a real-valuedfunction on [a,b] is Riemann-integrable, it is Lebesgue-integrable.

Iffn is a uniformly convergent sequence on [a,b] with limitf, then If a real-valuedfunction is monotone on the interval [a,b], it is Riemann-integrable.

Generalizations of the Riemann integral:

It is easy to extend the Riemann integral to functions with values in the Euclidean vectorspace for any n. The integral is defined by linearity; in other words, if, then. In particular,since the complex numbers are a real vector space, this allows the integration of complexvalued functions.

The Riemann integral is only defined on bounded intervals, and it does not extend well tounbounded intervals. The simplest possible extension is to define such an integral as alimit, in other words, as an improper integral. We could set:

Unfortunately, this does not work well. Translation invariance, the fact that the Riemannintegral of the function should not change if we move the function left or right, is lost.


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For example, letf(x) = 1 for allx > 0,f(0) = 0, andf(x) = 1 for allx < 0. Then, for allx.But if we shiftf(x) to the right by one unit to getf(x 1), we get for allx > 1.

Since this is unacceptable, we could try the definition: Then if we attempt to integrate thefunctionfabove, we get, because we take the limit as b tends to first. If we reverse the

order of the limits, then we get.This is also unacceptable, so we could require that the integral exists and gives the samevalue regardless of the order. Even this does not give us what we want, because theRiemann integral no longer commutes with uniform limits. For example, letfn(x) = 1 / non (0,n) and 0 everywhere else. For all n,. Butfn converges uniformly to zero, so theintegral of is zero. Consequently. Even though this is the correct value, it shows that themost important criterion for exchanging limits and (proper) integrals is false for improperintegrals. This makes the Riemann integral unworkable in applications.

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 Riemannintegral, 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 bounded Lebesgue-integrable functionfdefined on a bounded interval isRiemann-integrable if and only if the set of points wherefis discontinuous has Lebesguemeasure zero. An integral which is in fact a direct generalization of the Riemann integralis the Henstock-Kurzweil integral. Another way of generalizing the Riemann integral isto replace the factorsxi + 1 xi in the definition of a Riemann sum by something else;roughly speaking, this gives the interval of integration a different notion of length. This isthe approach taken by the Riemann-Stieltjes integral.

Viestruki integral:


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ACKNOWLEDGEMENT:Just as the definite integral of a positive function of one variable represents the area of theregion between the graph of the function and thex-axis, the double integral of a positivefunction of two variables represents the volume of the region between the surface definedby the function and the plane which contains its domain. (Note that the same volume canbe obtained via the triple integral the integral of a function in three variables ofthe constant functionf(x,y,z) = 1 over the above-mentioned region between the surfaceand the plane.) If there are more variables, a multiple integral will yield hypervolumes ofmulti-dimensional functions.

Multiple integration of a function in variables: over a domain ismost commonly represented by nesting integral signs in the reverse order of execution(the leftmost integral sign is computed last) proceeded by the function and integrandarguments in proper order (the rightmost argument is computed last). The domain ofintegration is either represented symbolically for every integrand over each integral sign,or is often abbreviated by a variable at the rightmost integral sign:

Since it is impossible to calculate the antiderivative of a function of more than onevariable, indefinite multiple integrals do not exist. Therefore all multiple integrals aredefinite integrals.


19/19

SCOPE:

Because gamma rays and X-rays cannot penetrate Earth's atmosphere, direct observationsmust be made from space. INTEGRAL was launched from Baikonur spaceport, inKazakhstan. The 2002 launch aboard a Proton-DM2 rocket achieved a 700 kmperigee.The onboard thrusters then raised the perigee out of the residual atmosphere, and theworst regions of the radiation belts. The apogee was trimmed with the thrusters tosynchronize with Earth's rotation, and thus, the satellite's ground stations.

INTEGRAL's operational orbit has a period of 72 hours, and has a high eccentricity, withperigee close to the Earth at 10,000 km, within the magnetospheric radiation belt.However, most of each orbit is spent outside this region, where scientific observationsmay take place. It reaches a furthest distance from Earth (apogee) of 153,000 km. Theapogee was placed in the northern hemisphere, to reduce time spent in damaging eclipses,and maximize contact time over the ground stations in the northern hemisphere. It iscontrolled from ESOC in Darmstadt, Germany, ESA's control centre, through groundstations in Belgium (Redu) and California (Goldstone).Fuel usage is within predictions.INTEGRAL has already exceeded its 2.2-year planned lifetime; barring mechanicalfailures, it should continue to function for six years or more. The spacecraft body("service module") is a copy of the XMM-Newton body. This saved development costsand simplified integration with infrastructure and ground facilities. (An adapter wasnecessary to mate with the different booster, though.) However, the denser instrumentsused for gamma rays and hard X-rays make INTEGRAL the heaviest scientific payloadever flown by ESA. The body is constructed largely of composites. Propulsion is by ahydrazine monopropellant system, containing 544 kg of fuel in four exposed tanks. Thetitanium tanks were charged with gas to 24bar(2.4 MPa) at 30 C, and have tankdiaphragms. Attitude control is via a star tracker, multiple Sun sensors, and multiplemomentum wheels. The dual solar arrays, spanning 16 meters when deployed andproducing 2.4 kW BoL, are backed up by dual nickel-cadmium battery sets. Theinstrument structure ("payload module") is also composite. A rigid base supports thedetector assemblies, and an H-shaped structure holds the coded masks approximately 4meters above their detectors. The payload module can be built and tested independentlyfrom the service module, reducing cost. Alenia Spazio was the spacecraft primecontractor.

integral [maths]

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