Infinitely Accurate Integrals?

[EquisDeXD]

So I know the general approximation formula using the limit of the function with n numbers of DeltaX times the height that changes over n intervals of delta x, but what if I want to be infinitely accurate or get the exact amount of area and not just some approximation? Would I use an infinitely small deltaX? How?

Edited October 7, 2012 by EquisDeXD

[studiot]

Well the simplest way is to stick to whole numbers.

[ajb]

Takelimits, very carefully.

[EquisDeXD]

Takelimits, very carefully.

 

But I don't see how you get infinite accuracy out of that, I'd have to pick some finite value of delta x to get the limit of it. I've seen it done before and it was with some kind of limit where you use some kind of summation formula for whatever type of function it is, but I can't find anything on google and I don't remember how it works, I remember seeing delta x substituted somewhere in the formula but I don't get how that achieves infinite or exact accuracy.

Edited October 7, 2012 by EquisDeXD

[studiot]

EquisDeXD

 

some kind of limit where you use some kind of summation formula

 

Well a good start would be to understand mathematical limits properly.

 

AJB has pointed you at a source for this, although I should allow Wikipedia is not always the most comprehensible.

 

You should thank him for that.

 

Did you want help understanding it?

 

 

[EquisDeXD]

Well a good start would be to understand mathematical limits properly.

 

AJB has pointed you at a source for this, although I should allow Wikipedia is not always the most comprehensible.

 

You should thank him for that.

 

Did you want help understanding it?

 

 

I get that I need to use limits, I just don't know how I need to use them to get an exact answer and I can't seem to find anything online.

[ajb]

Okay, to be much more accurate you should start with the Riemann integral, which was the first rigorous definition of the integral of a function on an interval. This is probably all you need to worry about at the moment.

 

You can think of the Riemann integral as the limit of a Riemann sum, such is what I was hinting at.

 

However, it is not actually sufficient for all purposes. This is remedied by Lebesgue integration.

 

More general notions of integration also exist in modern mathematics under the name of measure theory.

[ACUV]

I think you need to know when it matters and when it doesn't. It's all very well having the exact answer e.g. 3 , but is 2.999 good enough. Once I made an algorithm that calculated area under a curve and if I remember correctly the approximation had about 15 digits, 11 of which were after the decimal point before it lost accuracy in comparison to exact.

[EquisDeXD]

Okay, to be much more accurate you should start with the Riemann integral, which was the first rigorous definition of the integral of a function on an interval. This is probably all you need to worry about at the moment.

 

You can think of the Riemann integral as the limit of a Riemann sum, such is what I was hinting at.

 

However, it is not actually sufficient for all purposes. This is remedied by Lebesgue integration.

 

More general notions of integration also exist in modern mathematics under the name of measure theory.

 

I've heard of Lebesgue integration before, but I don't get what the point is exactly. Is it just for when you have more complicated functions so that you can switch between the two types?

 

 

<br style="font-size: 12px; ">I think you need to know when it matters and when it doesn't. It's all very well having the exact answer e.g. 3 , but is 2.999 good enough. Once I made an algorithm that calculated area under a curve and if I remember correctly the approximation had about 15 digits, 11 of which were after the decimal point before it lost accuracy in comparison to exact.

Oh yeah, this doesn't matter at all right now, I just want to know. Edited October 8, 2012 by EquisDeXD

[Bignose]

I think you need to know when it matters and when it doesn't. It's all very well having the exact answer e.g. 3 , but is 2.999 good enough. Once I made an algorithm that calculated area under a curve and if I remember correctly the approximation had about 15 digits, 11 of which were after the decimal point before it lost accuracy in comparison to exact.

 

I think this is a good point. Because unless it is a special case where an exact integral is known, truly infinite accuracy would require infinite calculations, clearly impossible to do in the lifetime of the universe. This is true of any iterative or limit calculation. I mean, any of those series approximations for pi can be done forever, but you'll never have 'infinite accuracy' for pi, since it is a never terminating, never repeating decimal representation, you'll never have the accuracy you seek.

 

For most needs, approximation of an integral can be done to sufficient accuracy within a finite amount of time. A lot of times properly choosing what constitutes sufficient accuracy is part of the art of solving the problem; and it usually involves balancing the uses of the available computing power & speed, and possibly the accuracy of any measuring equipment used.

[ajb]

I've heard of Lebesgue integration before, but I don't get what the point is exactly. Is it just for when you have more complicated functions so that you can switch between the two types?

 

The Lebesgue integral covers a wider class of functions than the Riemann integral. The Lebesgue integral and the Riemann integral are equal whenever the latter is defined. But, the Lebesgue integral is often well defined even if the corresponding Riemann integral is undefined.

[EquisDeXD]

I think this is a good point.

 

What? Who gives a something if I "need" to know it anyway? Who is anyone besides me to determine what I "need"? And what do you "need" anything for anyway? What if I was stranded on an island and I needed that math to calculate something but couldn't find out how it worked on google?

There's nothing wrong with just letting yourself starve to death, you don't "need" to eat food or drink water, it has no difference to the universe either way, but things do it anyway.

Edited October 9, 2012 by EquisDeXD

[Bignose]

What? Who gives a something if I "need" to know it anyway? Who is anyone besides me to determine what I "need"? And what do you "need" anything for anyway? What if I was stranded on an island and I needed that math to calculate something but couldn't find out how it worked on google?

There's nothing wrong with just letting yourself starve to death, you don't "need" to eat food or drink water, it has no difference to the universe either way, but things do it anyway.

 

Sheesh, was this meant to be really rude? It sure comes off that way.

 

You missed my and ACUV's bigger point that infinitely accurate just isn't going to happen except in some very special cases. That in the definition of your problem, you determine how accurate you need your calculations to be. There are some integrals that are only needed to be +- 5% accurate. There are others where 1 part per million is significant. The problem at hand determines how accurate one needs the solution to be.

[EquisDeXD]

Sheesh, was this meant to be really rude? It sure comes off that way.

 

You missed my and ACUV's bigger point that infinitely accurate just isn't going to happen except in some very special cases. That in the definition of your problem, you determine how accurate you need your calculations to be. There are some integrals that are only needed to be +- 5% accurate. There are others where 1 part per million is significant. The problem at hand determines how accurate one needs the solution to be.

 

This is what I'm talking about, with the exception of ajb, what are you guys doing on this thread if your not trying to answer the topic question? It doesn't matter if I need it or not, what matters is a scientific/mathematical question was asked so the scienceforums.net community should try to answer it as best they can as per the purpose of this website existing. If you can't/don't want to answer it, that's fine, I don't think any less of you, but if you don't want to answer it or help answer/ask any of the topic questions, there's really no point to be on this thread.

Edited October 9, 2012 by EquisDeXD

[Bignose]

This is what I'm talking about, with the exception of ajb, what are you guys doing on this thread if your not trying to answer the topic question? It doesn't matter if I need it or not, what matters is a scientific/mathematical question was asked so the scienceforums.net community should try to answer it as best they can as per the purpose of this website existing. If you can't/don't want to answer it, that's fine, I don't think any less of you, but if you don't want to answer it or help answer/ask any of the topic questions, there's really no point to be on this thread.

 

Do you think that maybe I am asking you these questions because maybe I am trying to help you answer the question? I'm not just asking them so I can hear the words rattle about in my head.

 

I am trying to lead you to clarify exactly what you mean.

 

If you are looking for truly infinite accuracy for all but the special case, then you might as well give up. Because it won't happen. It cannot happen in a finite amount of time. This is a perfectly valid answer to your question, whether you like the answer or not.

 

Now, if you took the time to think about the questions I asked you, about just how accurate you really need to be in your application, I can suggest numerous numerical integration methods. The techniques for them depend on things like -- how accurate you need to be, how quickly you need to compute answer, and how complex the integrand is. For example, a great deal of computational fluid dynamics simulaitons use very crude integral estimations at the beginning of a run so that the computations process quickly and return answers on the order of magnitude of what's correct. This allows the large scale flows to be resolved quickly. Then they transition to more accurate and costly computations in order to resolve the finer details of the flow. How fine details need to be resolved? Well, gee, there's that question about how accurate one really needs to be again!.

 

What I am not going to do is just write a treatise on numerical integration for you addressing all the issues, because quite frankly, there are many excellent texts written on the subject that apparently you didn't want to be bothered to read.

 

Lastly, if you have a problem with what people post in your thread, report it to a moderator. I was trying to help answer an ambiguous unclear question as best as I knew how -- by probing you for more details so I could best answer it. If you don't want this help, then feel free to go to another site. If you just want rote, unthinking, boiler-plate answers, then just stick to search engines.

[EquisDeXD]

Do you think that maybe I am asking you these questions because maybe I am trying to help you answer the question?

 

I think this is a good point. Because unless it is a special case where an exact integral is known, truly infinite accuracy would require infinite calculations, clearly impossible to do in the lifetime of the universe. This is true of any iterative or limit calculation. I mean, any of those series approximations for pi can be done forever, but you'll never have 'infinite accuracy' for pi, since it is a never terminating, never repeating decimal representation, you'll never have the accuracy you seek.

 

For most needs, approximation of an integral can be done to sufficient accuracy within a finite amount of time. A lot of times properly choosing what constitutes sufficient accuracy is part of the art of solving the problem; and it usually involves balancing the uses of the available computing power & speed, and possibly the accuracy of any measuring equipment used.

 

I don't see a question in there.

 

 

I am trying to lead you to clarify exactly what you mean.

 

If you are looking for truly infinite accuracy for all but the special case, then you might as well give up. Because it won't happen. It cannot happen in a finite amount of time. This is a perfectly valid answer to your question, whether you like the answer or not.

What I mean is I'm trying to get an exact number that perfectly matches the exact area under a curve which IS possible and I've seen it done in front of my own eyes before, but it was so long ago and it was somehow different from conventional integrals, in fact I think there was a specific symbol to represent infinitesimal, since technically you could achieve infinite accuracy by using an infinitely small delta x.

 

 

 

Lastly, if you have a problem with what people post in your thread, report it to a moderator. I was trying to help answer an ambiguous unclear question as best as I knew how -- by probing you for more details so I could best answer it. If you don't want this help, then feel free to go to another site. If you just want rote, unthinking, boiler-plate answers, then just stick to search engines.

Now your putting up a strawman, it's not the site that's a problem, it's you, your not helping the discussion in any way, your just dancing around an irrelevant problem. Who cares if I need it. If you look in the engineering section I posted how you make electricity from scrach if you were stuck on an island with every element. That's never going to happen, but regardless of that people answered it.

[Bignose]

What I mean is I'm trying to get an exact number that perfectly matches the exact area under a curve which IS possible and I've seen it done in front of my own eyes before, but it was so long ago and it was somehow different from conventional integrals, in fact I think there was a specific symbol to represent infinitesimal, since technically you could achieve infinite accuracy by using an infinitely small delta x.

 

And I am telling you that this will only happen in special circumstances. Most often, when there is a nice closed form for the limit when delta x goes to zero. In the great wide variety of integrals, ones where this happens is exceptionally rare. Most cases, you have to use numerical integration, where there will be no such thing as infinitely accurate. In the general case, infinite accuracy cannot happen.

 

(I am going to ignore the rest, if you have more problems with me, please take it out of the public viewing, and use the PM system.)

[uncool]

The Lebesgue integral covers a wider class of functions than the Riemann integral. The Lebesgue integral and the Riemann integral are equal whenever the latter is defined. But, the Lebesgue integral is often well defined even if the corresponding Riemann integral is undefined.

Well, that depends.

 

The Lebesgue integral covers a wider class on any set of finite measure. However, on sets of infinite measure, the Riemann integral can cover some functions that the Lebesgue integral can't. For example, Riemann integral covers

[math]\int_{1}^\infty \frac{1}{\lfloor x \rfloor} sin(\pi x) dx[/math]

while Lebesgue does not.

 

EquixDeXD, are you referring to taking the antiderivative? That is the only method I can think of that gives exact answers.

=Uncool-

Edited October 9, 2012 by uncool

[EquisDeXD]

And I am telling you that this will only happen in special circumstances. Most often, when there is a nice closed form for the limit when delta x goes to zero. In the great wide variety of integrals, ones where this happens is exceptionally rare. Most cases, you have to use numerical integration, where there will be no such thing as infinitely accurate. In the general case, infinite accuracy cannot happen.

 

(I am going to ignore the rest, if you have more problems with me, please take it out of the public viewing, and use the PM system.)

 

Well I can agree for something like sin(1/x) that after a certain point you can't really get a good integral because of the infinite number of sign changes, but even if you have some weird polynomial that goes all over the place can't you just break it down into separate limits where there's just one maximum or minimum and add them all up?

Edited October 10, 2012 by EquisDeXD

[Bignose]

Well I can agree for something like sin(1/x) that after a certain point you can't really get a good integral because of the infinite number of sign changes, but even if you have some weird polynomial that goes all over the place can't you just break it down into separate limits where there's just one maximum or minimum?

 

Polynomials are only one class of many, many different functions. You got all those trigonometric, exponential, logarithmic functions, most of which won't have a closed form integral. Then all those special functions, like the error function, gamma function, Bessel functions, hypergeometric functions, etc. There are an infinite number of functions. So again, it is really only rare special cases where closed form integrals will be found, necessitating numerical integration techniques the vast majority of the time, meaning that "infinite precision" will not be achieved in any finite amount of time for the vast, vast majority of integrals.

[ACUV]

EquisDeXD,

 

I have an opinion, I am entitled to make a post, I am on topic, if you want exact you may find it, you may not. The application of your answer is what is important. Do I truly care whether you do or don't get a correct answer, Why should I? I'm just trying to be helpful which is what the ethos of this forum is about.

 

If an approximation for an area of a field results in a farmer overspending on nutrients by an acceptable sum, there is no need to be exact.

[EquisDeXD]

Polynomials are only one class of many, many different functions. You got all those trigonometric, exponential, logarithmic functions, most of which won't have a closed form integral. Then all those special functions, like the error function, gamma function, Bessel functions, hypergeometric functions, etc. There are an infinite number of functions. So again, it is really only rare special cases where closed form integrals will be found, necessitating numerical integration techniques the vast majority of the time, meaning that "infinite precision" will not be achieved in any finite amount of time for the vast, vast majority of integrals.

 

Well if there's infinite functions then that means there's technically an infinite number of cases where infinitely accurate integrals are possible.

[studiot]

This is what I'm talking about, with the exception of ajb, what are you guys doing on this thread

 

I take exception to this since I was the first and only poster to give you a correct answer to your question as posed.

 

The only number system that is 'infinitely accurate' is the whole number system and I councilled you to stay within it.

 

If you do not it is physically impossible to achieve your stated goal.

Edited October 11, 2012 by studiot

[Bignose]

Well if there's infinite functions then that means there's technically an infinite number of cases where infinitely accurate integrals are possible.

 

ok, sure, that doesn't mean that isn't still rare. There are an infinite number of rational numbers, too, but there are more irrational numbers. Just because something is infinite, does not mean it isn't rare.

[EquisDeXD]

ok, sure, that doesn't mean that isn't still rare. There are an infinite number of rational numbers, too, but there are more irrational numbers. Just because something is infinite, does not mean it isn't rare.

 

Actually, that's exactly what infinite means, there are in infinite number of functions to find the exact integral of, infinity is larger than any actual number, so I don't think they are really too rare. I would think as a math expert that you wouldn't mistake infinity for being a real number and following normal algebra. Even if you try and fit infinite elements into a single set, the probability of any particular element being picked is 0.

Edited October 11, 2012 by EquisDeXD