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Topic: Improper integral


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  Improper integral - Wikipedia, the free encyclopedia
In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
In some cases, the integral from a to c is not even defined, because the integrals of the positive and negative parts of f(x) dx from a to c are both infinite, but nonetheless the limit may exist.
One can speak of the singularities of an improper integral, meaning those points of the extended real number line at which limits are used.
en.wikipedia.org /wiki/Improper_integral   (573 words)

  
 Integral - Wikipedia, the free encyclopedia
The integral between a and b of f(x) is the area between the curve y = f(x) and the x-axis in the interval [a, b].
Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite.
The Riemann-Stieltjes integral, an extension of the Riemann integral.
en.wikipedia.org /wiki/Integral   (1662 words)

  
 Integral - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-13)
Intuitively, the integral of a continuous, positive real-valued function f of one real variable x between a left endpoint a and a right endpoint b represents the area bounded by the lines x=a, x=b, the x-axis, and the curve defined by the graph of f.
An integral which can only be evaluated by considering it as the limit of integrals on successively larger and larger integrals is called an improper integral.
Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite.
encyclopedia.learnthis.info /i/in/integral.html   (1443 words)

  
 Improper integral: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-10-13)
In calculus, the integral of a function is a generalization of area_(geometry)area, mass, volume, sum, and total....
In mathematics, the integral of a function can be regarded in the simplest case as the area between the graph of that function and the x-axis....
The fourier transform, named after jean baptiste joseph fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions,...
www.absoluteastronomy.com /encyclopedia/i/im/improper_integral.htm   (1446 words)

  
 PlanetMath: improper integral
So there is no ambiguity in using the same simbol for improper integrals and usual Riemann integrals (but we will see that there is an ambiguity when dealing with Lebesgue integrals).
In particular this function is not summable (in the sense of Lebesgue integrals) on the interval
This is version 5 of improper integral, born on 2002-02-27, modified 2006-05-09.
planetmath.org /encyclopedia/ImproperIntegral.html   (248 words)

  
 Improper integral   (Site not responding. Last check: 2007-10-13)
In calculus an improper integral is the limit of a definite integral as an or both endpoints of the interval approaches a specified real number or ∞ or −∞.
One can speak of the singularities of an improper integral meaning those of the extended real number line at which limits are used.
By using the advanced Lebesgue integral rather than the Riemann integral one can in some cases bypass requirement but if one simply wants to the limit to a definite answer that fix may not necessarily help.
www.freeglossary.com /Improper_integral   (555 words)

  
 The Integral Document   (Site not responding. Last check: 2007-10-13)
An integral is applicable to a continuous function of an interval on a definite variable of the function.
In the illustration a and b are the limits of integration, the color lines are area elements, dx the lenght of the area elements and f(x) the height of the area element.
An improper integral can be convergent, in this case the result is a real number or divergent, when the result tends to infinite.
www.area48.com /integral   (910 words)

  
 Improper Integrals using active image map
Thus, the left-hand integral below is "proper" and describes the area of a triangle.
The right-hand integral is improper and describes an area (discussed in detail below) which extends indefinitely to the right.
One way to study improper integrals is to temporarily chop off an unbounded part of the area, find the size of the remaining bounded area, and then find the limit as the chop point is extended to include more and more area.
math.uww.edu /~mcfarlat/improper.htm   (338 words)

  
 notes2_10   (Site not responding. Last check: 2007-10-13)
A definite integral is called an improper integral if either there is an infinite limit(s) of integration or the function has a vertical asymptote in the interval over which you are integrating.
The other type if improper integral is one where the the function has a vertical asymptote in the interval of integration.
It is important to recognize improper integrals and to decide their convergence or divergence, especially when we use them to evaluate infinite series later in the course.
www-math.cudenver.edu /~rrosterm/notes2_10/notes2_10.html   (961 words)

  
 [No title]   (Site not responding. Last check: 2007-10-13)
The CPV is the limit (as B-->inf.) of the integral from -B to B of f(x).
Hence, the CPV of the doubly improper integral of f(x)=x exists because the integral from -B to B of f(x)=x is always zero (the neg.
The usual sense of the doubly improper integral is a more restrictive notion--notice that the CPV can give a finite value to the integral of a function which is unbounded as x-->+/- infinity, which may seem a bit strange.
math.niu.edu /~rusin/known-math/99/principal_val   (392 words)

  
 Improper integral   (Site not responding. Last check: 2007-10-13)
In calculus, an improper integral is the limit of a definite integral, as an endpoint,or both endpoints, of the interval approaches either a specified real number or ∞ or −∞.
By using the more advanced Lebesgue integral, rather than the Riemannintegral, one can in some cases bypass this requirement, but if one simply wants to evaluate the limit to a definite answer,that technical fix may not necessarily help.
It is more or less essential in the theoretical treatment for the Fourier transform, with pervasive use of integrals over the whole realline.
www.therfcc.org /improper-integral-85892.html   (353 words)

  
 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-13)
In both cases, the idea is to consider the improper integrals as limits that are symmetrically computed about the point of discontinuity (in the first case, the point of discontinuity is at infinity).
The key difference here is that the improper integral must satisfy a more strict criterion for convergence than the CPV, as required under the definition of integral as a Riemann sum.
Therefore, we want to maintain a sufficiently strict interpretation of the improper integral so that when it exists, it is independent of the choice of limits of integration.
www.mathforum.org /library/drmath/view/61246.html   (1016 words)

  
 MATH 155 SUPPLEMENTAL NOTES 17
Any integral that has infinite limits or points within the interval integration that causes the function to go to infinity are called improper integrals.
To evaluate this improper integral, we must divide the interval that is under consideration into two pieces.
This improper integral can be integrated directly, but let us use the limit comparison test to determine if it will converge or diverge.
faculty.eicc.edu /bwood/ma155supplemental/supplemental17.htm   (1239 words)

  
 Calculus II (Math 2414) - Integration Techniques - Improper Integrals   (Site not responding. Last check: 2007-10-13)
In this kind of integrals we are going to take a look at integrals that in which one or both of the limits of integration are infinity.  In these cases the interval of integration is said to be over an infinite interval.
At this point we’re done.  One of the integrals is divergent that means the integral that we were asked to look at is divergent.  We don’t even need to bother with the second integral.
In order for the integral in the example to be convergent we will need BOTH of these to be convergent.  If one or both are divergent then the whole integral will also be divergent.
tutorial.math.lamar.edu /AllBrowsers/2414/ImproperIntegrals.asp   (1415 words)

  
 Math Tutor Level 3 Text choice1=Integral choice2=Methods Survey choice3=Improper Integral
The first thing to remember is that any definite integral may be an improper integral - it is not always clear at the first sight.
Whenever we are given a definite integral to evaluate, we have to scan the integration interval for possible troubles.
It is therefore enough to know how to handle two basic types of improper integrals: those with trouble at the right endpoint and those with trouble at the left endpoint, with no other trouble present.
math.feld.cvut.cz /mt/txtd/4/txe3db4.htm   (1079 words)

  
 Appendix: Improper Integrals; Neuhauser, section 7.4   (Site not responding. Last check: 2007-10-13)
Such an integral is called an improper integral.
This demonstrates the phenomenon that the convergence of the integral is determined by how fast the graph approaches the x axis--in other words, how fast the function approaches 0.
It is not always easy and often laborious to directly verify the convergence or divergence of improper integrals using this direct method we have just illustrated.
www.math.jhu.edu /~js/Math107/coursenotes2/node11.html   (385 words)

  
 Improper Integrals
In this case, the integral converges and we may think of the area under the curve as being finite.
This is not too surprising since we don't expect there to be a finite amount of area under the curve.
In fact, this demonstrates the phenomenon that the convergence of the integral is determined by how fast the graph approaches the x axis---in other words, how fast the function approaches 0.
www.ugrad.math.ubc.ca /coursedoc/math101/notes/techniques/improper.html   (671 words)

  
 Mathwords: Improper Integral
A definite integral for which the integrand has a discontinuity between the bounds of integration, or which has ∞ and/or –∞ as a bound.
Improper integrals are evaluated using limits as shown below.
If the limit does not exist or is infinite, we say the integral diverges.
www.mathwords.com /i/improper_integral.htm   (68 words)

  
 6.7.1 Improper Integrals
Note that an improper integral is a finite number if it exists.
A non-basic-type improper integral will be broken into (ie, expressed as a sum of) basic types.
integral is broken into four basic-type improper integrals.
www.geocities.com /pkving4math2tor6/6_the_intgrl/6_07_01_improper_intgrl.htm   (673 words)

  
 154Syl.html   (Site not responding. Last check: 2007-10-13)
The integral test is established by showing that the appropriate integrals provide bounds from above and below for the partial sums of the series.
Since the partial sums form an increasing sequence, the theorem on monotone sequences implies that, if the improper integral is finite then the partial sums and hence the series converge.
And, if the improper integral is infinite, the partial sums increase without bound so the series diverges.
fas.rutgers.edu /home/beals/154/L15.html   (263 words)

  
 Math Tutor Level 3 Text choice1=Integral choice2=Theory choice3=Improper Integral
Since improper integrals are defined using a limit, most properties valid for definite (Riemann) integrals are still true.
The convergence or divergence of the above integrals cannot be influenced by the choice of the lower limit (as long as it is greater than zero).
There are functions that look similar and have convergent integrals both at 0 and at infinity; in other words, if we integrate them from 0 to infinity (which then must be done by splitting it into two integrals), it comes out convergent.
math.feld.cvut.cz /mt/txtd/4/txe3da4b.htm   (768 words)

  
 [No title]
The improper integral converges, therefore by the integral test the series converges.
The improper integral diverges, therefore by the integral test the series diverges.
Also, if you have not noticed, the integral test uses the concept of improper integrals to evaluate the resulting integrals.
faculty.eicc.edu /bwood/ma155supplemental/supplemental21.htm   (407 words)

  
 Improper Integral   (Site not responding. Last check: 2007-10-13)
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www.wikiverse.org /improper-integral   (498 words)

  
 Calculus II (Math 2414) - Integration Techniques - Comparison Test for Improper Integrals
We will give this test only for a sub-case of the infinite interval integral, however versions of the test exist for the other sub-cases of the infinite interval integrals as well as integrals with discontinuous integrands.
Let’s work a couple of example using the comparison test.  Note that all we’ll be able to do is determine the convergence of the integral.  We won’t be able to determine the value of the integrals and so won’t even bother with that.
So, it seems like it would be nice to have some idea as to whether the integral converges or diverges ahead of time so we will know whether we will need to look for a larger (and convergent) function or a smaller (and divergent) function.
tutorial.math.lamar.edu /AllBrowsers/2414/ImproperIntegralsCompTest.asp   (1531 words)

  
 Improper integral   (Site not responding. Last check: 2007-10-13)
It is recommended that the reader be familiar with antiderivative s, integral s, and limit s.
Such an integral can be evaluted by noting the antiderivative : arctan x.
These pathologies do not afflict "Lebesgue-integrable" functions, that is, functions the integrals of whose absolute value s are finite.
www.serebella.com /encyclopedia/article-Improper_integral.html   (820 words)

  
 Absolute Convergence of Improper Integrals
Since most of the tests of convergence for improper integrals are only valid for positive functions, it is legitimate to wonder what happens to improper integrals involving non positive functions.
As we mentioned before, this improper integral is not absolutely convergent.
Note that the integral is improper obviously because of
www.sosmath.com /calculus/improper/absconv/absconv.html   (360 words)

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