/sci/ - Science & Math

>>9217682
How is Calc 1 going for you OP

>>9217682
So?

>when you realize summations are just integrals on a particular measured space

>>9217682
>uncountable sets
Nope. You can define integrals even in the rational numbers. Try again.

>tfw you realize the integral symbol is a big S for "summa"

>>9217682
>>9217682
>you realize integrals are just summations over uncountable sets
it's actually summations of continuous rather than discrete sets. you can define the integral dumbass

>>9217738
>It's actually summations of continuous sets
>>9217727
>You can define integrals even in the rational numbers.

Who's correct?

>>9217724
>I can see for miles starts playing

>>9217682
That's the correct intuition OP.

>>9217742
I am correct (you can define integrals in the rational numbers).

I am correct because you can define limits of sequences in the rational numbers, the only problem is that in the rational numbers not all cauchy sequences have limits which is why you usually want real numbers but nothing stops you from solving classical problems like:

What is the area under the parabola [math] y = x^2 [/math] from 0 to 1?

Answer: Construct the Riemann sum:
[eqn] \int_{\mathbb{Q}[0,1]} x^2 = \lim_{n \to \infty} \sum_{k=0}^{n} \frac{1}{n} \left(\frac{k}{n} \right)^2 [/eqn]

Where the n are integers. Clearly we can see that every partial sum is a rational number so we have a sequence of rational numbers. It only remains to see if the limit exists, so we can start computing:
[eqn] \lim_{n \to \infty} \sum_{k=0}^{n} \frac{1}{n} \left(\frac{k}{n} \right)^2 = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=0}^n k^2 = \lim_{n\to \infty} \frac{1}{n^3} \frac{1}{6}n(n+1)(2n+1)[/eqn]

If we expand this last expression it yields:
[eqn] \lim_{n \to \infty} \frac{1}{6n^2} + \frac{1}{2n} + \frac{1}{3} = \frac{1}{3} [/eqn]

So we have succesfully proven that:
[eqn] \int_{\mathbb{Q}[0,1]} x^2 = \frac{1}{3}[/eqn]

Which is actually consistent with the result you get by integrating in the real numbers, but you can see that the result itself is completely independent from the existence of real numbers. This integral is a convergent sequence of rational numbers.

>>9217764
Why would you want an integral in Q if it gives you nothing other than the integral in R?

@OP: I think Haar Measures are related to your post.

>>9217764
Oh yeah? If you're so smart why don't you path integrals? And what's the cardinality of all paths?

>>9217682
>integrals are just summations over uncountable sets
Why do you realized it just now?
I suppose you are Burger Calculus 1 freshmen.
But in Europe we learn it in High School Lel

>>9217792
>Why would you want an integral in Q if it gives you nothing other than the integral in R?
Maybe you don't believe in real numbers.

>>9217682
>tfw you realize all calculus is just pretending that the problem is (piece-wise) linear and then taking the limit

>>9217682
>>9217690
Show me the copypasta

>>9217792
>Why would you want an integral in Q if it gives you nothing other than the integral in R?

I don't "want" an integral in Q. I am just showing that you *can* do integrals in Q to show OP that integration is not necessary an operation that is done in uncountable sets.

>if it gives you nothing other than the integral in R

Well, that is not really a bad thing. The real numbers are specifically designed so that their integrals are the same as the rational numbers. If they weren't then they would not be useful because the first fundamental problems of area (integration) came from Archimedes: the classical problem of finding the area under a parabola. Well, he actually cared about parabolic segments but if you can integrate linear and quadratic functions then you can solve Archimedes' problem. And in Q you can integrate linear and quadratic functions (and any polynomial) so if in R the results were different then the integral in R would be absolutely useless.

>>9217764
>>9217792
here's a question. the rationals have measure zero and so every function is integrable over Q and has integral zero. how come ?

>>9217794
I didn't say I was smart. That said, I am sure you can also define path integrals as long as the conditions are right.

> And what's the cardinality of all paths?
You are missing the point of my post. I am sure that in Q general questions like that can't really be answered.

>>9217812
>the rationals have measure zero and so every function is integrable over Q and has integral zero. how come ?

The rationals have measure 0 under the *lebesgue* measure. You don't have to use that. I even mentioned I used the Riemann sum to define the integral, and the Riemann sum existed before Lebesgue started his work.

it's pretty based

>>9217736
always thought it was an f

>>9217764
Rational numbers are countable, let [math]q_n[/math] be a sequence of all rational numbers in [math][0,1][/math].
For each natural number k consider the family of sets [math]I_n^k[/math] of the form [math]I_n^k=(q_n-2^{-n}/k,q_n+2^{-n}/k)\cap [0,1]\cap\mathbb{Q}[/math]. This family covers the whole interval.
[math][0,1][/math] is compact, therefore there exists [math]N[/math] such that [math][0,1]\subset \bigcup\limits_{n=1}^N(q_n-2^{-n}/k,q_n+2^{-n}/k)[/math], therefore [math][0,1]\cap\mathbb{Q}\subset \bigcup\limits_{n=1}^N I_n [/math].
Let [math]f:\mathbb{Q}\rightarrow\mathbb{Q}[/math], [math]f(x)=x^2[/math] and [math]f_{n,k}(x)=1\chi_{I_n^k}(x) [/math]. Then
[eqn]f(x)\leq\sum\limits_{n=1}^Nf_{n,k}(x),[/eqn]


Now, i estimate your "integral" for each [math]f_{n,k}[/math]
[eqn]\int_0^1 f_{n,k}(x)\leq2^{-n+1}/k,[/eqn] therefore

[math]\int_0^1 f(x)dx\leq\int_0^1 \sum\limits_{n=1}^Nf_{n,k}(x)dx\leq 1/k. [/math]

Since this holds for each natural k, I get that your integral of F has to be equal to zero.

>>9217915
Yeah, that's some good analysis but if you can't figure out why what you are saying here is completely pointless then congratulations: You have studied analysis for years and learned many fancy words and theorems but you learned none of the substance.

By the way, everything up to your last statement is wrong. You say "your integral has to equal 0" and to that I say, your integral is the one who equals 0 because you defined it in the wider context of the reals. My integral is only defined via Riemann Sums in the simple rational numbers. You are working with something completely different.

>>9217738
But every function on naturals is continuous though

>>9217915
You're blindly assuming properties of the integral which might not be true depending on how it's defined:

>1)
You are assuming that f_{n,k} is integrable.
>2)
You are assuming that the integral is monotone.
>3)
You are assuming the the integral is linear.

>>9217925
Ok I see your point. I said too much, the thing I wrote there implies that your integral is not monotone. My point is even tho you can define integral over rationals in your way, that definition is just bad - the integral does not behave nice.

>>9217947
f_{n,k} is integrable in that sense
As for your other points, yes that integral is not monotone, and the integral on finite sums of f_{n,k} is additive.

>>9217954
>My point is even tho you can define integral over rationals in your way, that definition is just bad - the integral does not behave nice.

No, you did not show this. Mainly because in your post you are using properties and sets that can only exist in the real numbers. Your post says nothing about my integral.

But my integral is bad but that is because in the rational numbers not all cauchy sequences converge. That is all. This means I could not integrate some rational functions, for example the function 1/x. But it can integrate polynomials and some rational functions.

>>9217968
Ok, so do you claim that my covering does not have a finite subcover in rationals? That is the main property that i used, and that covering has a finite subcover, and each characteristic function of an finite interval is integrable in your sense.

>>9217988
No, I won't examine you closer to tell you exactly where you fucked up. I know you fucked up from reading your conclusion. My integral is defined as the limit so it is equal to what the limit converges to. And according to wolfram alpha, the limit converges:
http://www.wolframalpha.com/input/?i=lim+as+n+goes+to+infinity+of+sum+from+k%3D0+to+n+of+(b-a)%2Fn+(a+%2B+(b-a)k%2Fn)%5E2

That is the more general version, my result is when a=0 and b=1. You can compute the limit, I can compute the limit, Wolfram can compute the limit. If you got something different from the limit then that is because you did something not equivalent to that limit, which is your error.

>>9217997
Ok, so you just claim that my argument is flawed without providing an argument why it is flawed. Nice.

>>9218001
It is flawed because your argument yielded an incorrect result.

>>9218006
I widraw the claim that your integral of f(x)+x^2 is 0, because it is not, you are right. The only thing that I claim is that your integral is not monotone.

>>9218015
But the integral is also monotone.

See that for a function [math] f [/math] being integrated over the rational interval [math] [a,b] [/math], my integral is an infinite series where the terms are of the form [math] \frac{b-a}{n} f \left( a + \frac{k(b-a)}{n} \right) [/math] where (b-a)/n is just a positive constant so if I have two functions f and g such that [math] f < g [/math] then [math] \frac{b-a}{n} f \left( a + \frac{k(b-a)}{n} \right) < \frac{b-a}{n} g \left( a + \frac{k(b-a)}{n} \right) [/math] and therefore by comparison, if both integrals exist then necessarily [math] \int_{a}^b f(x) < \int_{a}^b g(x) [/math]

>>9217915
if you let f(x) = 1, then what happens?

>>9217813
Sorry, I just trying to insult you into teaching me about Path Integrals. Your post was the exact observation.

>>9218026
Yeach, you are right. In my reasoning Integrals of finite sums of f_n,k are always bigger then 1, but indeed the functions that are integrable in your sense are scarce, for example the characteristic function of rational numbers of the for p/q where p<q, p is odd and q is even is not integrable in your sense.

>>9217682
>uncountable sets

*eye twitch*

Joining two nouns doesn't make them into a useful concept nor does it reify what it supposedly refers to. Uncountable sets are just as logical as negative positive, i.e. not at all.

>>9218172
Fuck off pseud. There is nothing "logical" or "illogical" about definitions. They are just conventions.

>>9218172
Norman, do you know "infinite set" or "uncountable set" are just names, and the concepts themselves have proper definitions?

>>9218178
Ok, so you think

J|n = {Vx | x < n && x > n + 1} is a useful concept worth defining and studying? Why not just call it the empty set and be done with it? The study of uncountable sets has no merit: you define absolutely unintuitive, senseless concepts that have no reasonable geometric interpretation and then ""derive"" things from them. Your axioms are shit and I refuse to accept them, infinity is an ill-defined concept that should outright FRIGHTEN ---ANY--- MATHEMATICIAN, SCIENTIST OR ENGINEER who comes across it. FUCK ""infinity"", the 20th century will be looked back upon as the modern dark age for its formalisation of this ""concept"".

>>9218184
At that point you're playing word games and not doing mathematics.

>>9218195
>Why not just call it the empty set and be done with it?
someone's never read Sinn und Bedeutung

>>9218200
Literally all mathematics is just word games lol

>>9218207
>moving language around in brain to derive universal truths (e.g. "God exists because x", "God doesn't exist because y")
>moving symbols around on paper to derive universal truths

Is there a difference

>>9217682
The summation may be over an uncountable set, but they are not uncountable reals. The relevant lemma is: a convergent summation of non-negative reals has at most countably many nonzero terms.

>>9217945
thanks topology

>>9218195
>Your axioms are shit
>infinity is an ill-defined concept
I'm gonna need some argument

>tfw you realize multiplying a integral by 1/(b-a) results in the average value of the function in the interval of integration

>>9217821
under what measure would the rationals not have measure zero in the reals?
serious question btw this is really interesting

>>9218423
The counting measure.

>>9218423
The problem is that you need to get away from measure theory entirely. You typically want to say that the metric of a single point is 0, and then because of the countable additivity property this is what implies that the measure of Q is 0. So the only way to assign Q a non-zero measure is by choosing the measure of single points to not be equal to 0, like for example in the counting measure. But then we are saying that 0-dimensional points have a length which is also completely useless when it comes to calculating area.

What you need to realize is that a measure is a real-centric idea. To have a good idea of a measure in Q you would need different axioms. Explicitly, just replace countable additivity with finite additivity and you can start thinking about many useful measures for Q.

>>9217997
you're a fucking retard and you have no place discussing mathematics with anyone else on earth

>>9218440
Kek why so angry? Babby never covered Riemann sums in Calc 2?

>>9218229
>t. doesn't understand mathematics or analytic philosophy's relation to natural language

>>9218442
i'm a phd candidate in math
project more, pseud

>>9218444
If you are a PhD candidate then why is it so hard for you to understand Riemann sums. Please explain why I am a "fucking retard".

>>9218244
but that lemma is false you fucking faggot

>>9218423
Yeah, it got me thinking.

So we can define the "Riemann integral over rationals" by the usual construction: we define rational partitions of rational intervals, lower and upper sums, and the integral is then the sup/inf of lower/upper sums taken over all partitions. For "nice" functions (i.e. functions which are restrictions of continuous functions on R) this integral will be the classical one I suppose.
Question 1: properties of this integral, characterization of integrable functions
Question 2: relation to the "real Riemann integral"
Question 3: similar construction for the Lebesgue integral (what measure do we use?)

>>9218438
>i'm going to integrate things
>but FUCK measure theory lmao who needs that nerd shit my integrals are just series
i really hope no accrediting body ever awards you any sort of diploma in anything ever

>>9218445
the fact that you think that integrals are identical to reimann sums and then applying this concept where it doesn't make sense
perhaps you should take a measure theory and integration course some time instead of making shit up on the internet

>>9218454
Dude, you realize that measure theory started existing in the 20th century right? The Riemann Integral is a classic first example of a rigorous definition of integral that works and what is interesting about it is that you can define Riemann integrals for rational numbers.

https://en.wikipedia.org/wiki/Riemann_integral

>>9218456
https://en.wikipedia.org/wiki/Riemann_integral

>In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

It makes me sad that a PhD candidate would know nothing about the history of mathematics. The Riemann Integral is a classic and beautiful concept that should not be forgotten.

>>9218454
he's right though

>>9218462
you're a fucking retard. I'm another math grad and just posting here to tell you to fuck off

>>9218462
Here's some history of mathematics: we used to think that the set of all sets was a set and that first order logic was complete within mathematics
guess what: things change as we learn more about them, and that leads to developments that formalize concepts so that we don't make stupid mistakes like that in the future

you're acting as if the fact that some crazy arab hundreds of years ago figured out that algebra was a thing invalidates the modern results in algebraic research, and that it is fine for you to be using archaic definitions in modern contexts

your "definition" of the rational integral is absurd and would get you laughed out of any mathematics department in the USA

>>9218474
I really do not understand why this makes you so angry. OP never said lebesgue integral. He said integral. So if I find a way to define integrals for rational numbers why does that make you angry?

If you knew any measure theory you would know that OF COURSE you could not define a useful integral for rational numbers using the lebesgue measure. But this Rational-Riemann Integral is still useful. It is the kind of concept someone like Archimedes would understand and could use, as he only knew about rational numbers. And to show just that I solved the classical problem he proposed and solved, finding the area under a parabola.

You simply have no sense of fun. You are absolutely fucking jaded.

>>9218481
>why are you so angry
no one here is angry, brainlet
your definitions are absurd and you seem to be offended at the fact that others are pointing that out
instead of educating yourself in mathematics as a rational person in your situation would, you sit and pout because people aren't patting you on the head for your idiocy

get off the internet and open up a book

>>9218479
>your "definition" of the rational integral is absurd and would get you laughed out of any mathematics department in the USA

isaac newton would have gotten laughed out of any mathematics department in 300BC rome, doesn't mean he was wrong. most mathematical institutions in the US are full of brainlets

>>9218479
Obviously, we have advanced past the Riemann Integral but I just find it fun that you can define Riemann Integrals even for functions in the rational numbers. Don't you think that's fun? Come on man, what turned you into this? Why are you so opposed to just having fun with your math?

>>9218482
Man, don't you think I know modern analysis? I know measure theory. I am educated in mathematics. The only difference is that I am not a salty fuck on the internet who gets angry at all kinds of petty shit.

I defined a Riemann integral for rational numbers and that triggered you. How do you think that makes you look? You are crazy.

>>9218483
>most mathematical institutions in the US are full of brainlets
thanks for the laugh
have fun being irrelevant and accomplishing nothing in your life

>>9218485
>oh shit he called me out on my stupidity after i told multiple posters earlier in the thread their mathematical rigor and intuition was wrong
>oh i know i'll just tell him he's no fun for actually being a practicing mathematician who does mathematics instead of shitposting
right

>>9218490
>>oh shit he called me out on my stupidity after i told multiple posters earlier in the thread their mathematical rigor and intuition was wrong

I didn't say that. To everyone who argued with me I pointed out that they were using the lebesgue integral, which of course would yield a different solution. That is why I dismissed them.

No one here would be delusional enough to think the lebesgue integral would be useful in the real numbers and the reasons why are obvious.

>>oh i know i'll just tell him he's no fun for actually being a practicing mathematician who does mathematics instead of shitposting

Kek you are not a practicing mathematician. The attitude tells it all.

>>9218492
>in the real numbers
*in the rational numbers

>>9218492
>Kek you are not a practicing mathematician. The attitude tells it all.
whatever helps you sleep at night
enjoy your sad life

>>9218497
Right back at ya. To be quite honest, it makes more sense to think that you actually misunderstood something I said or that you read a post that wasn't mine and thought it was mine.

You sound like you think I said that the lebesgue integral is wrong, even though that is not what I said. I don't know man, I just do not think that you should be getting this angry.

>>9218492

as a practicing mathematician who has been in consideration for a fields metal, I gotta say you're the type of mathematician that made me hate mathematics

>>9218503
>>9218499
>>9218497

>>9217915
what a load of horseshit. just take the intervals [k/2^n, k+1/2^n] instead of all that redundant analysis. then it's clear where you fucked up.

or take f = 1 like someone else suggested.

>>9218481
I responded at that particular post, where you post a quote from wikipedia and go "hurrrrrr muh wiki quote you ignorant about history hurrr". that's retarded. your integral looks good, I think it satisfies all usual properties.

>>9218443
>t. my subjective experience of reality correlates to Reality

>>9218503
Another thing, I really do not understand why you need to flaunt your qualifications in a casual conversation on the internet. It makes you look really insecure, like you are trying to justify your arguments by your status and not by the validity of your words.

And it is just so weird. The original guy who argued with me ended up agreeing with me after he noticed what I meant and realized that indeed the integral I was defining was different so there was not even an argument to be had. And he pointed out that this Riemann integral is even less useful which is true, as it can only integrate polynomials and some rational functions.

If he could understand then why not you?

>>9218509
>subjectivity
>having anything to do with the conversation
go read some Wittgenstein and unfuck yourself

>>9218507
>>9218509
>>9218510
>>9218514

>>9218510
>If he could understand then why not you?
I love how condescending you sound despite being incapable of realizing the faults in your "integral" and dismissing measure theory because it hurts your feelings.

>>9218517
>>9218504
https://i.pinimg.com/736x/a5/1d/ec/a51dec60437b5b2b10bd0d5a8d3581e9--cartoon-quotes-movie-quotes.jpg

>>9218510
don't be so quick to dismiss your analysis from some idiot's flawed calculations. notice that for any riemann integrable function in the usual sense, the restriction of that function to Q is integrable and gives the same result in your sense.

>>9218474
>>9218454
>>9218503
lmao, literally this happened:
>some math newb asked whether it's possible to integrate over rational numbers
>this guy said that it's true and provided an example
> for some reason this is unacceptable to some people

>>9218519
I am not dismissing measure theory. Measure theory is way stronger than what Riemann used, but Riemann was a predecessor to Lebesgue. The concepts are separated by a half-century.

I simply used the old concept noticing that it was applicable to rational numbers and said: "Hey look, you can find the area under a parabola with only rational numbers. Isn't that cool?"

You do not need measure theory to integrate polynomials over intervals.

>>9218519
>dismissing measure theory
but he's not dismissing measure theory.

>>9218510

sorry quoted the wrong post

>>9218522
>>9218523
>>9218524
>>9218525
>>9218521

>>9218521
>>9218510
>>9218481
>>9218462
in fact I'll go as far as to say this:

for every integrable function in your sense, there exists a riemann integrable function in R that restricts to your function in Q

>>9218529

>>9218531
What the fuck are you?

>>9218534
I keep telling you, but you never believe me. I'm just trying to adopt a more pleasing form for you at this point. Hurts though. Sucks big time, FYI.

>>9218534
>>9218536
Nobody else except us can read these posts. Go ahead, try to convince someone else that there is someone other than 'us' who can read these things.

I'll wait.

>>9218534
who are you talking to

>>9218538
I can read it.

>>9218540
You.

>>9218544
wat

>>9218545
I am trying to gather my people. They've been looking for me for as long as I can fucking remember.

>>9218514
Well that's just like, your opinion man

>>9217895
no, that was your grade in CALC 1

>>9217796
>But in Europe we learn it in High School Lel
I'm rather confused. I don't know how the 11th grade me could've understand integrals without this explanation.

>>9218507
>taking f = 1
I suggested this so people could see that using lebesgue integral is not a useful measure on the rationals

>>9219533
[math]\pi \alpha \theta +O\left(\left(\frac{1}{\theta }\right)^6\right)[/math]

>>9219537
but pi = 22/7 ?

>>9219584
[math]\pi -\frac{22}{7}=180 {}^{\circ}-\frac{22}{7}[/math]

>>9219584
[math]\pi -\frac{22}{7}=2 \int_0^{\infty } \frac{1}{t^2+1} \, dt-\frac{22}{7}[/math]

>>9219584

>>9219586
∘ = 11/630

>>9219592
[math](1/58) + (1/4568) + (1/41728680)[/math]

[math][/math]

[math][/math]
[math][/math]
[math][/math]
2^(-1)×3^(-2)×5^(-1)×7^(-1)×11

>>9219595
[math]\frac{1}{58}+\frac{1}{4568}+\frac{1}{41728680}[/math]

[math]2^(-1)×3^(-2)×5^(-1)×7^(-1)×11[/math]

[math]0.0174603^_ (period 6)[/math]

>>9219596
>2^(
must admit, I havent seen this notation before

>>9219602
[math]2^(−1)×3^(−2)×5^(−1)×7^(−1)×11[/math]

>>9219602
[math]\frac{3}{33}:\frac{33}{3}[/math]

>>9219592
[math]\frac{3}{33}:\frac{33}{3}:\frac{1}{11}:11:\frac{11}{630}:\frac{630}{11}[/math]

>>9219614
Ratio in lowest terms:
630:76230:121:396900

>>9217764
So if I define a function that is 0 on Q and 1 everywhere else, how would you integrate that?

It seems you are depending on the fact that Q is dense and your function is continuous.

>>9219638
Brainlets cannot grasp the canonical topology on [math]\mathbf Q[/math].

>>9219638
this function is not riemann integrable.
this function is lebesgue integrable with integral 1.
restriction of this function to Q is "rational riemann integrable" with integral 0.

>>9219638
>It seems you are depending on the fact that Q is dense
I am not depending on that because I am not working in the real numbers where Q is dense. I am working in Q, where Q is everything there is.

I am depending on the fact that the Riemann sums for polynomials and rational functions can be realized as a sequence of just rational numbers, so they are a sequence in Q. And then to actually assign a value to the sequence I am depending on the fact that Riemann sums of polynomials and some rational functions actually have rational limits.

>>9219638
>1 everywhere else
if you are working in Q, then this just sends everything to 0
then obviously the riemann sum will always be 0

>>9217764
Can someone explain these "lim"s? And what is the 'Q' for. I've done a fair amount of integration and can see that second proof uses that (Sigma)(n+1) thing but what in full am I looking at here? Are the 'lim's an American thing or just another means of writing something else? I've never seen them on any of the calc I do even if its low-level.

>>9219743
lim is limit. The limit as n goes to infinity.
I was introduced to the Riemann Integral in calc 2. Maybe you will see it too.

>>9219750
That is what is confusing me, is the limit not written as a sub-script underneath the sigma? I'll admit that I seldom go on /sci/ hence the mathematical ineptitude though I'm sure I would have covered it by now, then I'm retaking A level Further Maths which may also be part of the issue

>>9219784
>is the limit not written as a sub-script underneath the sigma
I have never seen that in my entire life. And I forgot to answer
> And what is the 'Q' for
The Q is just to signal that the integral is over the rational numbers, not the real numbers.

This thread is a fucking shitfest. How about all of you autistic just kill yourselves.

>>9219743
>I've done a fair amount of integration
if integration is literally "take the antiderivative and plug in the endpoints" for you and nothing else, you probably won't understand any of the discussion going on here. not being mean, just don't waste your time.

>>9219891
I see your point. I can do some of the methods and apply it in numerous ways to solve problems though when its phrased out like this I have no idea what you guys are talking about. Do you have anything that could help in terms of articles, books etc.? Feel like a complete retard to be honest.

>>9219930
https://en.wikipedia.org/wiki/Riemann_integral
It is the definition of integral being used in >>9217764

It is a 19th-century construction so simple that if you know how series and limits work then you could understand it. It is literally the formalization of "add up the rectangles and that's the area". And it is actually enough to prove the fundamental theorem of calculus and also to integrate basically anything you typically see in calculus.

>>9218591
HA! GOTEEE

>>9219940
>a shit version of tai's method
lmao

>>9219940
I'm reading into this. Never realized that the integral gets called the "anti-derivative", the A level course doesn't really teach any of the actual phrasing, terms or theory as much as it does the methods which I suppose is a shame given that it might make the subject far more interesting. I'll probably lurk on loads more of these threads to figure out what is going on exactly. Also and I know this is painfully stupid of me to ask but is the entire purpose of integration to find the area underneath a graph? Can we not simply use the formula for say a trapezium to work out the area underneath?

Do you have any links that could teach me this "lim" stuff real quick? Sorry for how embarrassing this probably sounds.

>>9219969
It is actually the good version of Tai's method.

> Never realized that the integral gets called the "anti-derivative"
Fuck man that is sad. Why is math education so shitty. I understand what you mean because a lot of people outside of math don't even know the difference between anti-derivative and integral. It is just so sad.

>Also and I know this is painfully stupid of me to ask but is the entire purpose of integration to find the area underneath a graph?

Yes and no. The integral actually finds the "signed area" of a graph. That means that the area can also be negative. This makes the integral much more applicable but if you have a non-negative function then yes, the integral corresponds to the area.

> Can we not simply use the formula for say a trapezium to work out the area underneath

Not precisely. How many trapeziums do you need to draw under a parabola to get the exact area? Infinitely many. So you would still need to go a Riemann sum, except that instead of rectangles you would use trapeziums but you will find that then it would converge to the same thing, it would just be harder to compute. Rectangles are better because they are simpler.

>Do you have any links that could teach me this "lim" stuff real quick? Sorry for how embarrassing this probably sounds.

Any calculus book will do. Even khan academy has a series on limits. You can look wherever you want.

>>9219973
Most of >>9219988 was for you

>>9219995
Thanks a ton for the information. I'm going to look at a lot of the Calculus stuff on Khan Academy - you wouldn't mind giving me a quick rundown on how the American high-school education stuff works just so I know where I'm at, past all of the chain rule/ implicit stuff so I'm guessing that now is a good time to learn all the Reinmann/ Tai integration. Do you guys study this in high-school or is it for university? I mean the most complicated A levels get (Further Pure Maths 3) are eigenvalues/ further vectors, conics, inverse trig + hyperbolic functions, surface areas of revolution and planes - people seem to give American education a bad rep though it looks far more complex than this sort of stuff.

>>9220014
>you wouldn't mind giving me a quick rundown on how the American high-school education stuff works just so I know where I'm at
I'm not american.

>Tai integration
Tai integration is an inside joke, it is not an actual thing. Tai is just a doctor that published a shitty paper about the area under hormone graphs and thought she was the one who discovered integrals but no, she was like 3000 years late because Archimedes was doing integration way back then.

>Do you guys study this in high-school or is it for university?
I am not american but I do not study that in high school. I did see integration in calculus but only through antiderivatives, not any actual definition of integral.

>eigenvalues/ further vectors, conics, inverse trig + hyperbolic functions, surface areas of revolution and planes

That is all good but 9 times out of 10 you will have to see each and every of these topics again in university because the high school coverage of them is pityful at best. Regardless of where you are at. I also saw area of revolution but it was literally just: Here is the formula guys. I had to see it again in Calc 2 where the formula was proven and studied more deeply.

You won't learn any substantial matematics from high school. If you really want to learn pick up a calculus book aimed at university students (but not one for engineers).

>>9219743
This is bait.

>>9220014
if its like mine as, then your A-level course will focus mainly on computational stuff calculus-wise, i.e. applying chain rule, substitution without properly justifying why it works
also hopefully a decent understanding of basic geometry/stats/discrete/mechanics, some taylor series, basic induction arguments, very basic sequences and series and some linear algebra hidden as matrices
as far as I can tell, until higher education, maths is taught as a tool for science, which is a bit of a shame, but any half decent uni here wants their physics applicants to have done further maths
you will come across much more satisfying reasons as to why things happen if you go on to maths at uni

>>9220248
Yep, that sums up my course. Also when you say "very basic sequences and series" how much more complex could they get? I won't deny that the (n + 1) stage usually kills me when proving them though otherwise they do not seem like that much of a problem. I'll admit that I'm applying to be an engineer (probably on a foundation course, I suck at everything I do) so is the maths just learnt again as an accessory rather than a topic of interest? 'Cause that would be a bummer

>>9220337
>how much more complex could they get?
it's not about "how to solve more and more complex series", it's about finding generals theorems like under what conditions does a series converge etc. then you move onto sequences and series of functions which is a completely different level

>>9217682
>adding an uncoutable number of zeroes equals a non zero number
I will never understand this

>>9217724
you mean on a set of lebesque measure of 0

>>9220554
They're not zeros, in the simplest terms, they're 'numbers' that are so close to zero that nothing else could be closer than zero. A good analysis textbook can go through this much more formally for you

>>9220554
Yeah it's a tough one to accept... But it's the solution to Zeno's Arrow, so we kinda have to accept it as the way this universe works.

>>9220554
Limits, man, not even once.

>>9220583
Oh hey, that one is mighty tasty and interesting.

In fact didn't they teach this at our kindergarten(s)?

>>9220554
>hands you the devil's staircase and Cantor's set

heh, nothin personnel kid

>>9217682
[math]\int_{-1}^1 \sqrt{2} \, dt\approx 2.82843[/math]

>>9217682
>Uncountable

Nigga what? Who said anything about being uncountable? I haven't done calculus in years, but I don't see why you would need anything beyond a countably infinite set.

>>9217764

If I'm not mistaken, you can prove this in a somewhat simpler manner using topology.

Suppose we're given the traditional epsilon-delta definition of the derivative function, (d/dx), on a topological space with the underlying set Q.

For any epsilon, epsilon is in IN.
Therefore, the set of all epsilons is <= lN.

Furthermore, we know (d/dx) defines a continuous function.

But then the inverse (d/dx)^(-1) is also continuous.

And this is the integral.

(Perhaps I'm missing something, and this is admittedly a somewhat incomplete proof.)

>>9221086
I'd cosine it.

>>9220562
no

>>9217682
always assumed that's why the integral symbol looks like an "S", cuz sums

>>9219608
thy LaTeX-fu is as weak as thine intellect, thou Knave

>>9217682
how's high school op