this makes no sense. how can you take a continuous sum of the

>Just high school things

>>7992546
It's just that i don't understand how this isn't an approximation.

>>7992544
makes perfect sense to me. learning about Reimann sums in calc I now

>>7992550
Limits my friend.

>>7992551
explain to me your understanding then if it truly makes sense.

>>7992544
>>7989498

>>7992550
you're approximating the area under the curve. the greater the number of "rectangles", the greater the approximation.


this is my basic understanding as a calc I student, probably somebody else here can give you a better explanation

>>7992544
So as it turns out there's a system of numbers that roughly corresponds to our experience of reality called the "real numbers".

What's more, it turns out that in this system of numbers, if a sequence approximations become closer and closer and to _each other_ then there is a unique real number that the approximations will approach as you get better and better approximations.

This property of the real numbers is called "Cauchy completeness". You can read more about it here:
https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

The unique real number that the approximations get closer and closer is the number we identify with the area under the curve.

The fact that we can talk about the "area" and the number that the approximations approach interchangeably and still have the rules of arithmetic make sense is non-trivial. This is why people spent time proving shit about calculus and real numbers.

>>7992550
What's the sum of this sequence:
[math]\frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, ...[/math]

It's 2. It's not approximately 2, it's exactly 2. Can you add up each element until you get to 2? No, because there are an infinite number of them. Can you prove that the sum is actually 2? Yes. This is just how calculus works. Just accept it.

>>7992564
so it's not necessary to get the exact value if you are aware that it's converging to a particular one due to the set of real numbers being cauchy complete?

>>7992554
damn i forgot power set. is that when it's the set that includes all subsets? does the set not automatically include the powerset?

>>7992573
but what if it doesn't converge to a rational number?

>>7992573
>This is just how calculus works. Just accept it.

This is a pretty ignorant and bullshit statement to make. The reason it works again has to do with the fact that real numbers form a complete metric space as stated here >>7992564

When you say that the sum is "exactly 2". What you're real saying is a statement about the fact that the series converges to 2 and you can replace the series with the number 2 whenever you encounter in some other expression and math will not break.

The question in the OP is valid and understanding why you can do the things you can do in calculus is just as or more important than manipulating symbols and learning clever integration tricks.

>>7992586
sorry for not understanding, but what do you mean by taking the least upper bound?

>>7992575
>so it's not necessary to get the exact value if you are aware that it's converging to a particular one due to the set of real numbers being cauchy complete?

I'm not sure what you mean by "exact value". The exact value IS the number the approximations converge to. This is the definition.

The fact that definition works is what is remarkable here (i.e. the integral will converge if your function is sufficiently well behaved, that there is only one value it converges to, that the value it converges to has properties that you would want the area under the curve to have). These are all the theorems of calculus.

>>7992592
but how do you know that it's tending to a particular value? what if it's diverging or cyclic?

>>7992591
The least upper bound of a set is the smallest possible number that is greater than every number in the set.

>>7992598

It DOES NOT always tend a a particular value. The fact that it does converge to a particular value for sufficiently well-behaved functions again is a result of the theorems of calculus.

Simple examples of things you can't integrate using Riemann integrals are:

f(x) = 0 if x is rational, 1 if x is irrational

or

f(x) = sin(1/x) between (0, 1)

>>7992544
You're approximating the area under the curve. It's not to supposed to be exactly the area but very close because you're taking the limits.

In addition you can get even closer if you make smaller and smaller rectangles. There's a point where this stops being useful though and seeing as you're taking the limits of function you're already pretty close.

>>7992616
I get the feeling that OP knows how it works already, but doesn't know why it works.

>>7992616
>You're approximating the area under the curve.
No you aren't, the solution to the integral is the exact area under the curve, not an approximation.

>>7992626
if you're talking bout Reimann sums, isn't that just an approximation?

>>7992599
so with an integral are you able to get an exact value for the area because the values of x that you summate with are from the least upper bound of the set of all potential values of x?

>>7992636

reimann sums is a *numerical method* for approximating integral problems

>>7992636
its just definitions dude.
its like saying .9999 = 1
kinda like saying "if I had a super long fucking stick that never melted I could poke the sun"
sure ok.

>>7992550
The main idea of calculus is that as your approximations get better, they approach the real area. So taking the limit of approximations gives you the real area.

>>7992643
>>7992642
so a definite integral, would find an exact area under the curve?

What I think OP is missing is that in algebra you can have values which are irrational such as pi() represented as a symbol. So when you say the area of a circle is pi()*r^2, you are not really calculating the area since it's irrational, but pi()*r^2 is still the exact area of that circle.

>>7992650
so how do you decide when there's enough precision to state that it's converging to the real area?

>>7992670
When you take the limit.

>>7992659
>definite integral

Man, I fucking hate the way they teach calculus in country that produces people like this. The only integrals are "definite integrals".

Stop confusing integrals with antideratives. The fact that the integration process is related to antideratives (the Fundamental Theorem of Calculus) is actually an amazing result that people don't appreciate just because of confusing terminology.

>>7992676
but how do you take a limit for a sum when you're trying to take infinite sums? for tangents on a line i get that you're trying to take two points and make them super close, but with sums you're making an infinite amount of them.

>>7992670
I think you can take better and better approximations and then take the limit, what number do they appear to be approaching? If you get 2 = 160, 4 = 168, 8 = 170, 16 = 170.5, 32 = 170.625, 64 = 170.65625,

then you could probably say that the area under the curve is 170.7

>>7992685
I'm taking this directly out of my calculus book - http://www.amazon.com/Calculus-MyMathLab-Pearson-Package-Edition/dp/0321963636/ref=dp_ob_title_bk

and yeah, they book does call anti-derivatives "indefinite integrals"

>>7992670

If you can completely integrate the equation then the precision is infinite

>>7992642
You literally don't have a clue what you are talking about. Integrals are defined by sums, not by antiderivatives (the riemann sum is a special case where your subintervals are evenly spaced which can be proven to work in a lot of cases). All of those nice antiderivatives you take in calculus class are allowed only because the Fundamental Theorem of Calculus is satisfied. When that fails, you either do integrals with the definition or whatever other theorem as been conjured.

>>7992694
>>7992685
The terms indefinite integral and antiderivative are used interchangeably. Both are the same thing. Obviously definite integrals are something else. Depending where you live, it might have been taught to you as the Riemann Integral.

>>7992705
Should anti-derivatives be taught only after introducing integrals then?

>>7992694
It probably isn't. An integral is not an antiderivative.

>>7992670
There is no "precision" involved. The limit of approximations is not itself an approximation. It is what is being approximated.

>>7992700
But what if you cant?

>>7992713
They should be and I'm pretty sure they were to me but it wasn't made distinct enough. First year calculus classes tend to handwave all of the theory and focus on the results.

>>7992713
Yes. In fact most semi-advanced calculus courses in colleges use this order:

1. real numbers
2. sequences, series, limits
3. (Riemann) integration
4. differentiation
5. fundamental theorem of calculus (only here do you learn that you can use antiderivatives to compute integrals)

This creates way less confusion about what integration is.

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You must understand that any function is itself the derivative for the area under a curve, for the function itself IS what defines the rate of change of the graph.

https://mymathsworld.wordpress.com/2010/11/10/integration-area-under-a-curve-proof/

As soon as you realize this, you can use anti-derivative techniques to find functions of areas under graphs. Those functions, too, could be integrated as well.

>>7992718
so the lim -->f(x) as x-->infinite is more of a method such that you gain an integral with an exact solution to it?

>>7992723
that might because you have actual mathematics majors lumped in with all the other students that have to take calc I for their major. For example the other day my calculus professor said he wouldn't even bother covering the mean value theorem because most of the students won't even take calc II.

[eqn] \int_{a}^{b} f(x) dx=\lim_{n\to\infty}\sum_{k=1}^{n} f(a+k\frac{b-a}{n})\frac{b-a}{n} [/eqn]

This is the limit you're taking.

>>7992726
what is that symbol called? dega?

>>7992734
That is exactly true. I never realized these things until I got to analysis and algebra. Man, I feel sorry for your prof. It sounds like he/she has given up.

>>7992688
That is what the fundamental theorem of calculus explains. If we have a function (the derivative) which describes the instantaneous rate of change of some function F, then F is the integral of the derivative. The integral is the reversal of the relationship between a function and its rate of change. If we can calculate derivatives with infinite precision we can also calculate integrals with infinite precision, because it's simply finding a function which F is a derivative of.

>>7992749
As a special case. It's not actually defined that way. It turns out it works and is useful but a lot of mathematical machinery goes into making that theorem.

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>>7992586
* Didn't mean to delete this post.

"I try to drew a picture to explain all the stuff I wrote in that post.

So as I defined, A Simple function maps to a finite set. So graphically we can think of each as a finite horizontal line.

Because each Simple Function is finite, we can define its integral as a finite sum.

So when we take an integral of a general function what we are doing is taking the least upper bound of the set of the integrals of all the simple functions which are less than f.

The integrals of the simple functions are the areas under each of the rectangles, so the integral of the entire function is simply the least upper bound of all those areas."

>>7992730
I don't understand what lim -->f(x) as x-->infinite is supposed to refer to. Finding the limit of a function as x tends towards infinity is separate from finding the integral.

>>7992751
Delta. It's lowercase delta

think of it like this.

You have a bunch of equally-spaced rectangles under this curve. The sum of their areas is f(n).
You'll agree that the more rectangles there are, the closer they will get to the actual area underneath the curve.

Calculus is not actually repeating this operation ad infinitum. It is simply observing where the horizontal asymptote of f(n) is, if that makes sense to you. It's just a trick to see where the function is going towards, even if it never gets there.

>>7992766
Yes. This is the Lebesgue integral though, and not what OP is asking about at all.

>>7992749
[eqn]
\int_a^b f(x) \; dx = \lim_{n \rightarrow \infty} \sum_{k=1}^{n} f \left( a+k \frac{b-a}{n} \right) \frac{b-a}{n}
[/eqn] Made it look brettier
>>7992751
δ = undercase delta.

>>7992756
yeah I think so. He's pretty damn intelligent tho. Has a PhD, I think his specialty is linear algebra. He usually teaches 300/400 level courses.

>>7992774
He is asking how you take the area under a curve. The Lebesgue Integral still does that, IMO in a more intuitive way.

>>7992544
Yeah it might take forever to get an exact number but you can get pretty close in finite time.
You can even tell how close you are by finding an upward bound

>>7992734
>>7992781

Fuck him. You should call him out for it. Tell him that he is creating another generation of students who don't understand calculus at all.

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>>7992685
>I have no idea what Im talking about and am retarded

>>7992685
> (the Fundamental Theorem of Calculus) is actually an amazing result

no its an obvious result souped up with confusing terms and archaic symbols

>>7992639
Pretty much, yeah.

>>7992790
Half the class are bubbly secondary education majors. Another quarter of the class are CS bros. Do you blame him? Any real math students probably already know most of the shit he is going on about.

>>7992721


You probably can, but would need the help of a mathematician, but in cases when you absolutely cannot, you can only approximate the solution. Keep in mind that a numerical solution is not an actual solution.

>>7992782
Yes, but this only true once you have the machinery so you can talk about norms and completeness in function spaces. This machinery is harder for beginners to understand than talking about convergence of sequences of real numbers.

>>7992793
You're an idiot and have no business calling other people retarded.

>>7992766
is there always a least upper bound?

>>7992550
iT'S A SCAM
actually the area under all of them is -1/12

>>7992814
i don't believe you for some reason.

>>7992814
No, you're thinking of
[math]\sum_{i=1}^{\infty} i=-\frac{1}{12}[/math]

>>7992796
I bet you think Stoke's Theorem is obvious too since it's basically the same principle. I'm not as smart as you I guess where such things are obvious to me.

>>7992812
If the function is integrable, then yes.

Part of the definition for a general integrable function is that [math] \int {\left| f \right|\operatorname{d} \mu } < \infty [/math] .

>>7992825
are there definitions for general integrable functions that don't rely on the need for a least upper bound, or is that where numerical methods like reimmanian sums are utilized?

>>7992829
A function can be just locally integrable. So for instance while a function may diverge at some point on the full domain, you can still work with it on some subset of the domain.

>>7992860
can there be a function where there's no subset of the domain which is locally integrable?

>>7992870
Yes, but you can not integrate it.

For instance for a function to by integrable, it must be measurable. (in the sense described here: http://boards.4chan.org/sci/thread/7989422#p7989498)

So if a function is nowhere measurable, there is no way it could be integrable.
Also it should be noted, every Riemann Integrable function is Lebesgue Integrable. But not the other way around.

>>7992823
I actually did...

>>7992705
>the riemann sum is a special case where your subintervals are evenly spaced

Wrong, when partitioning an interval it does not have to be the case that x_{i+1}-x_i are the same for all i.

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Let's take integration to the instantaneous level.

Let's say that the area under the curve [math]f(x)[/math] from [math]a[/math] to [math]b[/math] is [math]A_{a}^{b}[/math]. That's our notation.

Now, let's add on the next bit of the curve, or in other words, integrate from [math]a[/math] to [math]b + dx[/math]. Well, the whole area is going to be the original area ([math]A[/math]) plus the little bit that we added on, right? And when that [math]dx[/math] is really small, that added area can be represented by a rectangle with base [math]dx[/math] and height [math]f(b)[/math]. So, if we rearrange things, we get:
[eqn]A_{a}^{b+dx} - A_{a}^{b} = f(b)dx[/eqn]
[eqn]\frac{A_{a}^{b+dx} - A_{a}^{b}}{dx} = f(b)[/eqn].

This looks an awful lot like differentiation, doesn't it? As a matter of fact, this says that the function can be represented by the rate of change of area. This is the fundamental theorem of calculus, and it's why integration is essentially anti-differentiation.

>>7992544
How can you get from one place to another?
Wouldn't it just be a never ending operation of getting halfway to your destination, and then halfway from your current position to the destination, and then halfway from your current position to the destination, and then...

>>7992885
To add on, there are functions which are neither Riemann Integrable or Lebesgue Integrable. But are defined in terms of an Improper Riemann Integral.

Ex. https://en.wikipedia.org/wiki/Improper_integral#Improper_Riemann_integrals_and_Lebesgue_integrals

>>7992917
holy shit you actually have a point. how can i travel through space if i can divide it an infinite amount of times such that each finite element is nonzero?

>>7992550
It IS an approximation, like anything dealing with curved lines

>>7992726
This is only true for a very limited class (also the least interesting class) of functions.

>>7992913
(If the function is continuous at b)

>>7992929
good point, continuity is critical for integral exact solutions

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>mfw people actually gave me constructive insight

wew

>>7992942
Only for the Riemann integral.

>>7992921
Because, clearly, space is not composed of an infinite number of elements

>>7992942
Also wrong.
It's just the case that continuity at a point is necessary to apply that Fundamental Theorem of Calculus at that point.

The integral is exact for a great many functions that are continuous nowhere on a compact set.

>>7992947
isn't space a smooth continuous manifold?

>>7992947
Or space is a hilbert space.
It's one or the other.

>>7992952
what's a hilbert space?

>>7992955
An inner product space (has some notion of geometry (it makes sense to define angles and lengths)) that is complete.

To be complete, it has to be the case that when I move by smaller and smaller increments (so small that for any number you name, I can tell you the number of the step after which all my steps are smaller in length than the number you named) I end up somewhere in that space.

>>7992952
A vector space can not describe our physical space. As far as current physics is concerned, our space is a manifold.

Unfortunately the types of manifolds required for QFT are not the same as the ones required for GR.

>>7992962
is that like a power set of all reals?

>>7992972
The real numbers are a hilbert space with the inner product (a,b) = a*b.

>>7992965
I think we're safely in the classical regime when talking about human motion.
Insofar as QFT and GR disagree, I think it's pretty reasonable to agree that we go about our daily lives in a hilbert space (although QFT and GR may apply heavily to some of the things we use and observe).

>>7992980
There are a such thing as "Quantum" Manifolds. More accurately Non-Commutative Manifolds. Using non-commutative geometry you can actually unify GR with the Standard Model. However there is a small yet extremely important issue, we only currently know how to define Non-Commutative Riemannian Manifolds. But because of time, we need pseudo-riemannian manifolds.

(Time is something you obviously can't account for on a vector space.)

this whole thread is like an attempt to teach a calc 1 student analysis, and I can't even tell if that's a good thing. yes, the handwaving explanation of limits and the lack of understanding how the field of real numbers actually works SHOULD make the inquisitive student skeptical and ask for rigorous statements. but, sometimes those students are actually asking much less deep and much more misinformed questions, and then when you give a rigorous answer it's taken the wrong way.

>>7992965
loop quantum gravity would like to have a word with you

>inb4 a worse meme than string theory

>>7992550

just read apostol or rudin fag bcuz u prolly heard a gay meme explanation telling u its a sum when kind of its not sooo yeahhh lolllll fug off

>>7993200
>>7993322
Heyyy hahaha double dubs double buddies lmfao

>>7993200
>loop quantum gravity would like to have a word with you
>inb4 a worse meme than string theory

Loop Quantum Gravity is complete bullshit. At least String Theory looked promising for awhile there.

>>7992544
If you wanted the real answer much of the time it would be
It is an approximation.
All of the numerical methods for integration of anything beyond second order polynomials are approximations with error

>>7994083
but integrals themselves aren't approximations are they? you can get the exact value for the area by utilizing a limit because you're endless operation of obtaining more precision, from what i read in this thread, converges if the function has a least upper bound, and with that, it has a general integral.

>>7992544
how can you plot the graph of the function in the first place when you cannot evaluate the value of the function as the independent variable increases by an infinitesimal amount?
You can find f(0) and f(1) quickly, but to know the shape in between, you need f(0.5), and then you'll need f(0.25), and much later in the process you'll need f(0.0009765625), and much much later the independent variable is so small your computer can't even produce it.

Integration is not a special case of this "never ending" phenomena of continuous quantities.

>>7994133
Yes, he's confusing methods to numerically compute the integral with the definition.

If you need to compute
[math]\int_3^7 x^2[/math]
which is defined as a limit to infinite of a sum with more and more terms, then you can show in analysis that that limit is
[math] (7^3-3^3)/3 [/math]
and this is an exact value.

If you apply a numerical method, you'd evaluate x^2 at a large but finite number of terms and the value depends on the grid you use in the approximation.
You can't optain the value that is the limit by the numerical method for arbitrary very small grid size.
But that doesn't mean the integral is an approximation.
Limits are discustingly non-constructive, though.
fuck LEM

>>7992951

only in mathland

>>7994188
on a side note, does pi converge to an irrational number? or is the fact that it's irrational a divergent?

>>7992544
>wouldn't this be a never ending operation?
Only if you're dumb enough to iterate finitely.

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posting in epic thread

>>7993677
he is the hater of LQG

>>7992544
You get a function, not a value. Only a couple of function families have a finite area under their infinite length.

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Lamination for mod;

Walkin' up to the club like: