Can someone explain to me intuitively what exactly is going on when we integrate?
I understand that it's used for finding the areas under graphs, and I know the method, but what am I actually computing? With differentiation, we're finding the gradient at specific points using a line which has a difference of 0 between our point and another arbitrary point. I don't, however, understand what integration actually is.
>>7989422
Youre summing infinitely many rectangles with height H and width lim x --> inf for 1/x
>>7989422
You are taking the sum of "thin slivers" under the function. You then take the limit of the width of those slivers as they go to an infinitesimal width, dx or whatever.
Its just essentially an antiderivative. If you know derivatives you can understand integrals. Id you take physics youll see it clearly
Ok, Let [math]M[/math] be a set.
A subset of the powerset of M , [math]\sigma \subseteq P\left( M \right)[/math], is called a sigma algebra of M if the following properties hold:
1. [math] M \in \sigma [/math]
2. [math] A \in \sigma [/math] implies [math] M\backslash A \in \sigma [/math]
3. [math]{A_1},{A_2},... \in \sigma [/math] then [math]\bigcup\limits_{n \geqslant 1} {{A_n}} \in \sigma [/math]
[math] \left( {M,\sigma } \right) [/math] is called a Measurable Space.
A measure is a map [math] \mu :\sigma \to \bar {\mathbb{R}}_0^ + [/math] such that the following conditions hold:
1. [math] \mu \left( \emptyset \right) = 0 [/math]
2. [math] {A_1},{A_2},... \in \sigma [/math] such that [math] {A_i} \cap {A_j} = \emptyset [/math] then [math] \mu \left( {\bigcup\limits_{n \geqslant 1} {{A_n}} } \right) = \sum\limits_{n \geqslant 1} {\mu \left( {{A_n}} \right)} [/math]
[math]\left( {M,\sigma ,\mu } \right)[/math] is called a Measure Space.
A function [math] f:\left( {M,{\sigma _M}} \right) \to \left( {N,{\sigma _N}} \right) [/math] is called measurable iff for all [math] A \in {\sigma _N} [/math] we have [math] {f^{ - 1}}\left( A \right) \in {\sigma _M} [/math]
A measurable function [math]S:M \to \mathbb{R}_0^ + [/math] is called Simple if it maps to a finite set. i.e. [math] S\left( M \right) = \left\{ {{s_1},...,{s_n}} \right\} [/math].
Finally, let [math] f:M \to \bar {\mathbb{R}} [/math] be a Non-Negative Measurable function. (Non-Negative just means [math] f\left( m \right) \geqslant 0 [/math] for all m.)
Then we define the integral as:
[math] \int {f\operatorname{d} \mu } \equiv \sup \left\{ {\sum\limits_{z \in S\left( M \right)} {z \cdot \mu \left( {{S^{ - 1}}\left( {\left\{ z \right\}} \right)} \right)|S = simple\& S \leqslant f} } \right\} [/math]
This is often also written in the form [math] \int {f\operatorname{d} \mu } = \int {f\left( x \right)\mu \left( {\operatorname{dx} } \right)} [/math]
>>7989498
fucking analysis and measure theory
is there no end
>>7989498
That was interesting, though not exactly intuitive!
(I'm still trying to make sense of that last part)
Anyway my answer for OP was that it is a sum.
>>7989523
Pretty much what you are doing is looking at all the simple functions below f, and then summing up the areas under all those simple functions.
>>7989545
Oh and then you have to take the supremum (least upper bound) of them.
>>7989422
Think about a function and differentiation. It is increasing at a rate based on the slope at a given point.
For integration, instead of a rectangle, imagine a trapezoid with end points of x and x + delta x and f(x) and f(x + delta x). As delta x approaches zero the slope between f(x + delta x) and f(x) approaches the derivative or slope of the function (i.e. the derivative of the function).
So integration (area under the curve) is implicitly entangled with the slope (i.e. derivative).
The fact that area under the curve is directly related to the antiderivative is not necessarily obvious but ultimately is the result.
>>7989592
I personally think the definition of the Lebesgue Integral is more intuitive than the Riemann Integral. It seems like a much more solid construction.
>>7989422
Looking at how numerical integration works shows you the underlying process. You are adding up chunks of area underneath the curve. There are gaps because there are a finite number of chunks. As the number of chunks is pushed to infinity, the gaps become infinitely small and the exact area under of the curve is the result of the sum of all the (now infinitesimally small) chunks).
The symbology itself tells you what's occurring. The ∫ symbol, originating with Leibniz, is a long 'S' for sum and what your are summing up is the "dx" infinitesimals.
(I now await the math majors coming to tell me how that's right in the general sense but completely wrong in the particulars.)
>>7989600
nice bait
just kidding
that's pretty shitty bait
>>7989614
Also, the "chunks" you add up do not necessarily need to be rectangles. For integrals of solids of revolution, the "chunks" are disks, as shown in common illustrations for the process.
>>7989634
I'm serious. Maybe its because I have always hated undergrad level analysis but loved topology.
The definition of something like a measurable space is a lot like that of a topological space. (infact sometimes the sigma algebra can be generated by the topology).
>>7989614
A better animation of numerical integration.
>>7989743
If that's not intuitive to you then you will never practice math.
>>7989781
>if you haven't learned what i learned two semester ago you are nothing
>>7989498
You think that he fucking understands any of this? He's most likely taking Calculus 1 bro
>>7989498
>sup
not much you?
>>7989422
It might be easier to think in discrete time. What integration is is adding up every sample and multiplying by the time step, dt.
>>7989820
I define everything you need to know.
The only "prerequisites" to the definition I gave are:
- knowing basic set operations
- knowing what a function is
- knowing what a least upper bound is
>>7989831
>knowing basic set operations
Not formally covered before calc
>knowing what a function is
Not covered formally before calc
>knowing what a least upper bound is
Not covered at all before calc
Congrats your an idiot and a shitty teacher
>>7989743
>The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.
>>7989498
I know less now.
>>7989841
>Functions aren't formally covered before Calc
You fuckin' what.
>>7989877
I can't get past the name of the school.
Most schools use Stewart and say fuck all about the formal definition of a function
Can you post the page that formally defines functions?
>>7989813
kek I love this putdown gonna have to steal this some time
>>7989422
the fundamental theorem of calculus is really just telescoping series. Telescoping series are easy to undo because they have pieces that cancel each other out. So for example, if I ask you to add up all these numbers:
(2-1)+(3-2)+(4-3)
You should see that
-1 + 4
is all that's left at the end points. Except the distance between points is smaller and there are more of these little difference terms.
>>7989841
In what type of shitty class do you start learning what an integral is before learning basic set operations, functions, and basic properties of the reals.
>>7989907
Not one in pure math?
Most likely like OP.
I studied it in real analysis, but pretty sure most people study calc by rote. Thats why Stewart has such a nice mansion
>>7989911
I still don't get how you can do integration without knowing the definitions of a function and a real number (requires the notion of a l.u.b.). Both of which require knowledge of basic set theory.
>>7989592
>not some abstract nonsense like algebraic geometry
sheer autism
>>7989918
All of algebraic geometry isn't abstract nonsense. Varieties are fairly intuitive. Schemes are where the nonsense starts.
>>7989884
Here's the entire chapter that just introduces functions. Please tell me how much more formal it needs to get than this.
>>7989917
> whats the integral of x^2
Idk lets apply fundamental theorem of calc
> x^3/3 + C
Your acting like the average college student can prove fundamental theorems of integration from definitions of Riemann, Darboux or Stieltjes integrals.
I couldn't and I aced calc.
Are you a math student? Or did you study out of the US?
>>7989932
Yeah that's what I thought.
A more formal definition is, a set of ordered pairs, such that if (a,b) and (c,d) is in the set and a=c, then b=d
https://en.wikipedia.org/wiki/Function_(mathematics)#Definition
>>7989943
Oh, that's the magic phrase you're looking for. Well fine, it's introduced even earlier.
And the original point of all of this was that you have a good understanding of functions before you step into Calc, so I don't even know where this is going.
Should've appended the page before it for good measure.
And a good understanding of the LUB property?
I sure didn't see it...ever as an undergraduate
You're creating a function F(x) whose value at x is equal to the area under f(x) from 0 to x.
>>7990041
>I sure didn't see it...ever as an undergraduate
How?
>>7989422
Start with Riemann Sums and the mean value theorem.
Careful observation will note that Riemann sums can be manipulated into a telescoping series by utilizing the mean value theorem.
This will look like this section, albeit in reverse (bottom to top of the section).
https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Proof_of_the_second_part
>>7989422
It's a continuous sum.
>>7990084
I'm not a math student?
Supremums actually gave me hell when I started with analysis.
Am I alone? Physics students? Engineering students? Did you study sups and LUBs?
>>7990319
I did, but I generally don't use it at work
t. IE
>>7990345
Yes and I'm from a third-world country, so I doubt you have any idea where I studied. But I will tell you that we skimmed over sups and LUBs in calc I and II.
>>7990362
No. Even Riemann sums are pretty disgusting to look at.
Get a decent analysis book and start cracking.
>>7989422
There's two ways to measure the height of a staircase.
You can add up the height of each individual stair.
Or you can add up the total height of the final stair and subtract it from the total height of the initial stair.
The fundamental theorem of calculus states that these two techniques will give you the same answer
>>7989498
>hurr look how smart i am
>>7989498
Why would you ever think that this would be an acceptable response to someone just starting to mull over the basics of calculus?
Are you really trying to help OP understand, or are you trying to confuse him so you will look smarter?
Language should always be used to enlighten, never to confuse. You should tailor your language to your audience, which in this case is probably a first year Calculus student.
As Einstein once said: "If you can't explain it simply, you don't understand it well enough."
>>7989422
OP, I would encourage you to look up the Riemann sums used in calculus.
There are only two that we know of that can be factored exactly. Every other integral is solved either through substitution, manipulation or approximation.
First is the Geometric Series:
>http://mathworld.wolfram.com/GeometricSeries.html
Second is the Telescoping Sum:
>(can't find it, should be in the back of your textbook)
These are used to solve the integral of an exponential, and a polynomial respectively. Once you've worked through these proofs and understand them, you should have a much firmer grasp on the nature of the integral.
For some reason these are not emphasized in class, although they absolutely should be
>>7989498
This is nice. Now you can look at integral in constructive predicative logic.
http://arxiv.org/abs/0808.1522
>>7989422
Say you're driving a constant speed v for an hour. Your distance travelled is then v * one hour.
Now say you drove at half speed for the first half hour. Then your travelled distance is:
(half an hour * v/2 + half an hour*v).
Now say your speed varies constantly during this hour, then your distance travelled is the integral of your speed with limits 0 and 1 hour.
It's the area under the graph
/thread
>>7989422
Differentiation is like >_<
Integration is like <_>
>>7989422
You're creating a very large number of rectangles under the function-curve, and summing up the area. You know this, right?
It computes the area between the function and the x-axis. It's actually a bit more complicated than that, but that intuition will serve you well for Calc classes.
>It's just the area under a curve guise!
>>7990555
> I did an undergrad course in functional analysis and I'm the fucking boss of maths. Screw you, Gauss!
Well first you get your sigma algebra...
>>7989592
>modern analysis is the most autistic
>not topology
Topology is pretty fucking autistic.
>>7990597
Well to be honest, in my undergraduate fa class we didn't do spectral measures/integrals.
That pic is from my postgrad spectral theory script.
>>7990597
How is functional analysis an undergrad class?
At my school you have to go through both the Undergrad Real Analysis and Grad Real Analysis sequences before you can take Functional Analysis.
I've withdrawn from calc 2 at my community college already once, and I think I have to again. Do I even bother trying again or should I just hero myself?
>>7990974
Not him but if you make it through the 3 undergraduate real analysis courses you can take functional analysis here. It's a split 4th year/graduate course taught to both, so the lectures are the same but the graduate students have to do a researchy project on top of what the undergrads do, and sometimes they get harder assignment/test questions, although I don't know if functional analysis would even have any exams.
>>7991037
What gave you so much trouble?
>>7991037
I did twice, except I was majorly depressed, not attending class, and literally did not attend tests. Once i 'got better' i managed to convince the department to let me just take classes that had it as a prereq and I've been fine since. Famalama, just keep progressing. Don't give up if Calc 2 is what you need to go on with your goals.
>>7989422
>given differentiable function f(x)
>let g = df/dx
>intuition: at point a, f is about to change to the "next" value at the rate of g(a)
This means if you know g(x) and at least one point on its integral f(x) (say f(a)), then you know the rest of f(x) by finding out the "next" value after f(a), and the value after that, and so on by looking at the respective g values. That's how i'd explain integration intuitively, even if it'd have some technical errors.
>>7990797
> high schoolers can handle measure theory
>>7992161
The difficulty is arbitrary. A higher schooler could definitely handle basic measure theory if the school had offered a "Intro to Formal Math" type of course (i.e. logic, basic set theory, numbers, functions, etc.)
>>7992178
this is sort of like saying you could become a concert trombonist if you just practiced for 2 hours every day instead of jerking off to shemale foot porn
>>7992207
Not really. If you had studied sets and functions, and understood what you studied, then there is no reason you should have a significant problem understanding what a sigma algebra or a measure is. The definitions are fairly simple.
>>7992267
I just can't read it because iPhone doesn't do java
>>7989614
Then how do you take an integral with bounds -infinity and +infinity?
>>7989821
underrated
>>7989433
fpbp
It's a refined riemann sum.
>>7992325
Take the sum over all real numbers rather than a finite interval. But really, you should know that improper integrals are a limit of integrals as the bounds go to infinity, in which case it's obvious how that gels with what he posted.
>>7990362
This is the Lebesgue integral. For pedagogical purposes we should first make sure we're comfy with the Riemann integral.
In the Riemann integral we begin by partitioning the domain of a function into subintervals. Multiply the length of each subinterval by the one of the functions values on that subinterval. If one of our subintervals in [0, 2] then we this step might correspond to evaluating 2*f(1). Then we add all these guys up and we get an approximation of the integral. Keep doing this with successively smaller subinterval lengths, and you'll end up with the integral.
Sometimes this doesn't work, and functions which we really really want to be integrable just aren't. So here's the hack that Lebesgue introduced: partition the range, not the domain. Then multiply each element in the range by the size (or measure) of the set that gets mapped to that element. So, if [0, 1] and [4, 5] both get mapped to f(1), then we evaluate 2*f(1). Then sum all these up.
If it seems like we're doing the exact same thing that we do with the Riemann integral, then congratulations, you've got some intuition. And, in fact, the Lebesgue integral is equal to the Riemann integral for every function a flesh-and-blood human being could ever picture in his head. Moreover, if a function is Riemann integrable then its also Lebesgue integrable. But, as mathematicians, we should always try to push the boundary on the limits of autism and come up with obscure functions that only exist on paper. Some of those functions are not Riemann integrable, but they are Lebesgue integrable.
That's what's going on here. If you wanna think about how the words coincide with the notation, then the only important line in his retarded elitist response is the second to last line. The rest of it just sets the stage. Specifically what the rest of his post is doing is defining "measurable functions". Then you can integrate measurable functions.
>>7992417
This is the analysts way of throwing out every awful function (which tend to be so obscure that not even mathematicians want to think about them) and so that their beloved definition of the Lebesgue integral won't be tainted by said awful functions stubborn unwillingness to be integrated. That's right: we're segregating functions here. Good functions go under the integral, bad functions stay the fuck away. Also, see pic related for a visualization of the contrast between the Riemann and the Lebesgue integral
>>7992325
>>7992429
can you explain what types of functions can be integrated this way but not by riemann integrals?
how is this different than a riemann integral along the y axis?
>>7992325
Take the limit of each end going towards and away from zero and see if it converges towards a value or diverges towards infinity.
>>7992791
The function [math] {\chi _\mathbb{Q}}\left( p \right) = \left\{ \begin{gathered}
1,p \in \mathbb{Q} \hfill \\
0,p \notin \mathbb{Q} \hfill \\
\end{gathered} \right. [/math] is not Riemann Integrable on [math]{\left[ {0,1} \right]}[/math].
But [math]\int\limits_{\left[ {0,1} \right]} {{\chi _\mathbb{Q}}\operatorname{d} \lambda } = \lambda \left( {\mathbb{Q} \cap \left[ {0,1} \right]} \right) = 0[/math]
where [math]\lambda[/math] is the lebesgue measure.
Its just the anti derivative ffs.so if the antiderivative is mibus 2x the integral is 2x
>>7992992
Not just stupid, but wrong.
>>7992791
Yeah, but tomorrow (amerifat time zone memes). Too drunk right now, and also I'm on mobile.
>>7992847
Alright, notationFag, we're all impressed. But you could just say "zero on the irrationals and one on the rationals".
>>7989498
>ITT freshmen and stupid sophomores who have never taken functional analysis
>>7993043
http://mathworld.wolfram.com/CharacteristicFunction.html
>Given a subset A of a larger set, the characteristic function chi_A, sometimes also called the indicator function, is the function defined to be identically one on A, and is zero elsewhere.
>>7993053
I'm aware. But if you're trying to /actually/ explain something to someone then you should choose the simplest path available
>>7990444
holy fuck this is some dank shit thanks senpai
>>7989846
Every time
>>7993343
this one is for compact regular locales and takes values in the dedekind reals (positive or negative).
there is another theory of integrals, simpler,
https://www.cs.bham.ac.uk/~sjv/integration.pdf
where this time, you do not require compact regular locales as a domain, but any locale can be used and valued through some valuation.
the trick is that now your valuations-measures are only valued in the lower reals.
As said, they use the rieman integral and the choquet integral.
The two notions coincide when they overlap.
>>7992417
>But, as mathematicians, we should always try to push the boundary on the limits of autism and come up with obscure functions that only exist on paper.
fucking amazing.
>>7993031
maybe not; he hasn't formally defined "mibus"
>>7993490
kek
The fucntion is the height. dx is the width. The integral symbol means summation of all the rectangles :^)
>>7993719
>The integral symbol means summation of all the rectangles :^)
Define this. How do you sum uncountably many objects?
>>7989846
False equivalence
>>7989498
>>7993747
what is a series
>>7992417
If the Lebesgue integral is so autistically rigorous that it can even integrate functions that don't exist then why is the Gamma function un-integrable?
>>7994069
Good question. How do you sum a series over an uncountably infinite set?
![tITBK[1].png](https://i.imgur.com/58A0UiH.png "tITBK[1].png")
can someone explain the intuition behind integrating differential forms ? the classical riemann and lebesgue integral are intuitive as fuck, but I don't know how to visualize the integration of k-forms (or how to visualize k-forms in the first place). to me it seems like okay let's see what happens when we just remove the wedges, oh it works like a charm so let's define it this way.
>>7994131
ftp://ftp.cis.upenn.edu/pub/cis610/public_html/diffgeom4.pdf
>>7994096
by being slick. for the geometric and telescoping series.
>>7994076
Hm? The Gamma function is analytic in the complex plane everywhere except at z = {0, -1, -2, ...} so I'd suspect it's integrable.
>>7994151
Both the geometric and telescoping series are taken over a countably infinite set. I asked about an uncountable one.
>>7992791
>how is this different than a riemann integral along the y axis
Think about the function f(x) = x^2. That function divides the unit square [0, 1] x [0, 1] into two sections, and the section about f(x) has more area than the section below f(x). Ignoring the fact that the function in >>7992417
isn't well-defined as a function of y then there's no way our integral along the y-axis would be able "capture" the fact that a set of measure 2 gets mapped to f(1).
>can you explain what types of functions can be integrated this way but not by riemann integrals?
Not really. The indicator function here >>7992847 is a classic example. If you use the Riemann integral, then with some types of partitions the integral gets arbitrarily close to 0 but with other types of partitions it gets arbitrarily close to 1. If you use the Lebesgue integral then there's only one way to partition the range, namely 0 and 1. The computation ends up being 1*(measure of the rationals) + 0*(measure of the the irrationals) = 0 + 0 = 0. But, I hesitate to make any generalizations based on this one example.
You know how an MRI let's you see a slice of whatever you are scanning at small intervals, thus giving you a sort of image of the whole thing? When we integrated we stack together all those slices.
