You ask for intuition about volume of cone, then say you don't want any of the usual explanations.

Do you know what an integral is?

It is reasonably easy to show combinatorially that $\sum_{k=0}^n \binom{k}{r}=\binom{n+1}{r+1}$This also follows from Pascal's Rule. This is related to counting the number of lattice points inside the standard $r$-simplex with side $n$. Is this the kind of explanation you are looking for?

@coffeemath, I believe, I explained well why they don't work for me.

@MichaelMorrow, for sure I know. See my post more carefully. Why are you asking? Any useful input based on integral?

@Kapil, where from these equation come? I mean, how does is relate to the question and come from it? How and why do you jump to Pascal rule and combinatorics? Can you show the pathway? Can you please provide some references I could start with?

Since the derivative of $\frac{x^{n+1}}{n+1}$ is $x^n$, the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$. You can think of the integral as an "antiderivative". This is why the factor of $n+1$ shows up in the denominator. It doesn't just "come out of the blue".

@DamirTenishev about Kapil I don't know what simplexes are but here's their Wiki article: en.wikipedia.org/wiki/Simplex . Something about choose functions (the combinatorial thing) if you scroll down.

@MichaelMorrow, have you read my question??? This was exactly the origin of the question - why we have this in the integral. Please read carefully from the words "First of all, where from did I come to this question." You explanation is a classic example of "because this is written in the book". My question is why in the integral we have n+1 in the denominator. Well, the explanation "integration is inverse operation to differentiation so we should have n+1 in denominator" helps, but it doesn't answer to the question "where from it comes". My question is about nature, your answer is about book.

@MichaelMorrow, if you can explain "Why 3" or "Where from 3 comes" in a different way than "because we have such a formula" or because (I like this explanation as an intermediate) "integration is inverse operation to differentiation so we should have n+1 in denominator", I will be very thankful for it. Just explain this numbers in logical way. My question doesn't come out of the blue, I want to understand integrals better and key word here "understand"; just "such a formula" hardly helps here. Please help, if you can.

@PineappleFish, thank you, I already found this link, looking into it and its relation to the problem.

Well, you can also view integration as area under a curve. So the integral $\int_0^1 xdx$ represents the area of the triangle formed by the line $y=x$, the $x$-axis, and the line $x=1$. If you graph this, visually the area should be $\frac{1}{2}$. So, when calculating the integral, we get $\int_0^1 xdx=\left[\frac{x^2}{2}\right]_0^1=\frac{1^2}{2}-\frac{0^2}{2}=\frac{1}{2}$. So you see that if the factor of $2$ was not in the denominator, the area would not be correct. So here, the factor of $2$ comes because the area of the triangle is half the area of the square.

It's kinda funny this reminds me of this talk youtube.com/watch?v=k7MuXCOlE6M specifically from 25:50 to 30:37. The factor of 1/2 in areas or 1/3 in integration is relatively common. I realize that doesn't answer anything, I just thought it was cool. So about the "deeper meaning", maybe it has to do with integration. But as you know integration isn't just about formulas.

it's weird though, you don't always get "3's" when finding volumes in general. There is a good reason that three's come into play with cone-type things in particular...

@MichaelMorrow, absolutely and I started (my investigation) with this. And it is easy to explain, since we have an area of triangle, so $1/2$ of the box. This is true and understandable. The question is why this goes to higher powers like 3, 4, 5. One case (integral of x) is not the prove for all such cases. How to make (prove) the generalization or make the deduction for all powers? Otherwise, somebody could say, "well for all other powers the denominator then will be 2".

The fact that integration is the inverse of integration is also not at all trivial; I mean, if you tried to explain the relationship of slopes to areas to someone who's never taken calculus it'd seem downright bizarre.

@PineappleFish, yes this is very close to the way I came to my question here. It seems that I found something close to the answer here: math.stackexchange.com/questions/860071/…, see the accepted answer. It seems that Pappus's centroid theorem could help me, except the comment I just posted over there - it works for an external axis. Anyway, it is close.

@PineappleFish, on the relation between integration and differentiation. Yes, you are right, this is in the way of easy visual explanation, but it helps me with the original question about the integral. Of course, the first answer I found for myself is what MichaelMorrow said here "we must fit the reversing operation". It really helps, since it is very easy to explain the multiplier in differentiation visually. Helps, but doesn't explain. My problem is deeper. I need to invent my custom numerical integration technique for a specific task, so I need to understand it inside out...

Lots of facts in math just don't have simple intuitive explanations. I wouldn't expect there to be any simpler explanation than the ones that you've already seen.

so there's this: youtube.com/watch?v=S0_qX4VJhMQ&t=11s . I think this gets at a fundamental part of the differential aspect; because a square is two dimensional, you'd expect 2's to come out in front when you differentiate it. Also this is related to binomial coefficients and pascal's triangle - only the second term, the n term, stays when you differentiate, whenever you have an n-dimensional box, although n-dimensional boxes are pretty hard to visualize past n=3. Therefore finding the area under a curve manually using integration might be a better route if you want to...

understand why it works for the general n-dimensional case.

Perhaps n-dimensional (-referring to dimensions of space) is where you'll apply integration, perhaps you view integration as an area under a curve, an inverse of differentiation, or something else. It's actually pretty thought provoking, since there are multiple ways to understand why an integral being equal to something is true. It's kind of philosophical: what is integration? But as long as you can realize all of these problems where integration is used as equivalent, then you're all gucci.

@EricWofsey Maybe, but simplicity means different things to different people too, and sometimes the person asking doesn't know what's considered simple, either, or at least they don't know it until they see it.

@DamirTenishev a Riemann sum would work for finding the area under the curve..

Although, the algebra gets kinda messy for finding the exact value with a Riemann sum. If you buy the chain rule, then logarithmic differentiation is another option for differentiating $x^n$. Just ask me about it. Also, I can't figure out how to make it move the discussion to a chat.

A way to see the factor that applies to both the derivative and integral might be to look at the volume of an $n$-hypercube with a vertex fixed in space. For the derivative, if the side length increases a bit, then most of the volume increase occurs through the $n$ opposing $(n-1)$-dimensional faces moving outwards by that amount, resulting in $\Delta x^n \approx n x^{n-1} \Delta x$.

For the integral, if we start with a tiny $(n+1)$-hypercube and accumulate larger and larger faces in only one direction, that results in $\int x^n$. And if we accumulate in all $n+1$ directions, it forms a larger $(n+1)$-hypercube, so $x^{n+1} = (n+1)\int x^n$. (The resulting picture is actually the cutting/partition into pyramids from the linked question.)

@DamirTenishev btw, what kind of 'custom numeric integration' are you talking about? Is it like integrating a Taylor series? That's all I can bring to mind right now. I'm afraid I'm probably not much good with numerical computer type stuff, but if it's simple enough I might be able to help. In any case, it sounds pretty cool!

duplicate of math.stackexchange.com/questions/623/…

@epimorphic I couldn't have said it any better myself. Also Damir; ya might wanna check out this too: math.stackexchange.com/questions/3328140/…

@amWhy It's true that the question is partly almost an exact duplicate of what you mentioned, but they also gave at least two reasons why they weren't satisfied with the answers there. What is the policy on trying to ask a question again (although, it isn't completely identical) if you weren't happy with the answers? I remember a meta post about it but can't find it anymore. Also the XY problem ( meta.stackexchange.com/questions/66377/what-is-the-xy-proble‌​m ): I think this user might be...

asking the question because they are trying to get at something else too, and marking it as a duplicate would not be a productive way to help this user.

then the user, @Pineapple, needs to say as much. This site is not in the business of trying to mind read a user's post.

@everyone This comment thread is rapidly moving off-topic. It is up to the asker to clearly phrase their question, and indicate their confusion. Let us please not speculate on what they are thinking, and allow them some space to respond to the numerous issues raised here.

@amWhy the question isn't mathematically clear yes, but it is clear enough as an anchor to further help the user get a satisfactory answer. There isn't a single clear line, but if we keep at I believe it can be productive for the user. I see no issues with working with the user in order to help them tease out the question, or more accurately, some question to which they found the answer useful. It's not mindreading, because when the OP responds then we can know their mind for certain.

@XanderHenderson I am not trying to speculate what they're thinking. I'm just providing information which may be useful to them in both learning and clarifying the question. I agree that they need time and space to respond. It also seems a tad insulting to the user to indicate that they haven't done a good job of trying to indicate what they've been thinking regarding the question, as they have clearly given ample information and after they have demonstrated...

an amount of research effort above and beyond the minimum.

@PineappleFish "the question isn't mathematically clear yes." This is all that needs to be said. The question isn't clear. The procedure is to close the question so that the asker can make it clear.

After they have clarified the question, it can be reopened. This is, in large part, meant to prevent answerers from providing off-topic or irrelevant answers before the question can be made clear. Closure is not punitive.

@XanderHenderson except that nobody has instructed them on how to make it clear. Closing the question "as unclear" doesn't encourage clearness (in fact I mainly think it encourages people to ask worse questions, because the thought police have forbid people from asking questions that they personally feel are good).

"After they have clarified the question, it can be reopened." bruh you haven't even talked to them and attempted to help them formulate an actually useful question in the future. So what is closing it going to do exactly??

No, I cannot pinpoint what exactly is not clear to me---the question you linked to in your opening seems to completely resolve the question you are asking about. I fundamentally don't understand what you are asking which hasn't already been addressed. I have no idea what kind of answer would satisfy your query.

@PineappleFish, regarding the video "Derivative formulas through geometry". This is exactly what I meant when I said that it is easy to explain why we have this multiplier for differentiation; it is nicely shown in the video. If I had such visual way to explain the denominator in the integral I would be happy.

@PineappleFish, thank you for the thoughtful answer below the link to the video. I have to watch all these materials before replying back. I will do soon when I watch and read the sources you mentioned.

@epimorphic, thanks for the input. It requires some drawing to catch. Anyway, your explanation (as I see it now, maybe I should read more carefully) don't take into account the specifics of the form of all these bodies (like I mentioned in the question) - we have something related to the topology here, I guess. Let me think about your approach a little bit.

@PineappleFish, on the "custom numerical integration" I mean that I have a task with some specific changes in the variables and I can work out some custom integration method which could work better than famous Runge-Rutta 4 or some other of dozens numerical integration methods. For example, if you are doing N-Body system simulation you know the laws that affect the bodies and you can develop better numerical integration based on the system to be simulated.

@amWhy, no it is not and I explained this well in the beginning of the question. Their question is "Why?" and any explanation works for them from accepted far-stretched 3 prisms to sum of powers of two. These answers don't work for me, since they don't explain the nature of this division. They prove, yes. They work for 99% of people, yes. But they don't work for me, since my question is different: what is the origin of this.

@XanderHenderson, please see the video youtube.com/watch?v=S0_qX4VJhMQ. It explains why we have multiplier for the derivative and it is intuitive. It is understandable. I can see where this 2 or 3 comes from. I can work with this.

Now see the situation for the integral. All answers are really far-stretched: "I see that there are three these things or that things", but no one of them could explain clearly and easy: why do we have this 3 in denominator. My question is crystal clear for me: what is exactly three here? Three pieces of what do we have?

I want something I could come to, let's say 12 years old boy and say "Well, the integral is one power higher because we are summarizing by the same variable, so line becomes a box or box becomes a cube and we have a divisor with the same value as a power because..." and in this "because" I don't want to draw far-fetched sophisticated images or go deep in math of series. I want the same simple explanation as I have for the differential.

Would it help if I update the question with this?

Is the purpose of MSE to help human beings answer their questions, even if it means extra elbow grease for the person answering it, or provide fun problems for answerers that have clear solutions? Also, self studying anything independently is extremely hard in part because you don't yet know what a good question to ask is, or the good questions might not even cross your mind.

Without any kind of personalized feedback and too broad a scope the good/appropriate questions just float right past you; just by someone taking the time to address you personally, it ensures you that you are going down the right path, and I guess this is one of the reasons why I come here to ask questions - 'help' in math and understanding math, is not always simply about the answering of a 'technical' proposition.

Anyways, I've been here for hours though so I'm leaving for a bit to chill because I'm starting to lose it and go cross-eyed.

@DamirTenishev No. I am not going to watch a 20 minute YouTube video in order to understand your question. It is your job, as the asker of the question, to phrase your question in as succinct and clear a manner as possible. No one should have to watch 20 minutes of video to get your point.

@DamirTenishev Why do you believe that such a simple explanation exists? What could possibly be more simple than "decompose a cube into three pyramids, then apply Cavalieri's principle to generalize to the cone." I don't understand how this answer fails to answer your question. Note, also, that a nearly identical argument applies to higher dimensional examples, though it might be easier to work with simpler figures, e.g. standard $n$-simplies.