What is Integration? 3 Ways to Interpret Integrals 

Thank you!!

@@MathTheWorld I learned integration in 1970 aged 14. I was only taught the third way: the tiny bits version. I was not polluted by the first more common two. At university, I discovered this ‘tiny bits’ version was how we were all taught. Now I understand the difficulty of taking outcomes from small samples, but my subjective impression was that everybody my age in the U.K. was taught this way. So, my subjective opinion is that something changed along the way to my kids (now in their thirties and one using integration routinely at work) who were taught (thankfully) all three. That the ‘little bits’ way of teaching this is not perfect is hardly the point. Teaching a system so that it can be utilised is one thing: it means the user can make use of integration in real life. That integration has strengths, weaknesses and value in some situations and not others is another thing to learn about integration but is a different (if related) topic to how it is most constructively taught. So, a piece on why this change occurred in teaching after the early 1970’s would be interesting and could be used to illustrate the strengths and weaknesses in the real world.

The claim of this video is a click bait. When they say you learned it wrong, they don't know who their target audience is, so they must have some people in mind when affirming this. I wasn't going to watch, but the claim was so hard to miss that I wanted to know if there's a correct way to learn integration. But I learned all three ways, starting from tiny bits summing up to a whole.

That sounds like a bot, or those bad actors from 4 am commercials

​@@Coolcmsc and what are they learning now at 14? Basic factoring? Something has fallen off.

Thank you for sharing! We're glad to hear there are classes out there that focus on this very important conceptualization of integration!

I agree this main idea of this video: integration is adding up many tiny bits after applying some function. I want to mention the multivariable perspective, where integration as adding up "tiny bits" of quantities (generalized density) over some region S. Density can be thought of as a function f: S --> R. When this quantity is constant, (ie. 1), then we have a region of constant density, so we have a one-to-one map between volume and the result. Thus, we find the generalized volume of the region. I argue that fundamentally, integration is still about finding *generalized* area, or volume. However, I think that the idea of integration finding the "area under the curve" is dissatisfying, because it forces us to think of a function as strictly a graph. This is a limitation, because not all functions are conveniently interpreted as graphs. For example, a 3d density map from R^3 --> R cannot be drawn as a graph (in 3D). I think the more powerful approach is to use the region S and density f approach. From this perspective, the "area under a curve" can be thought of as the area of the region S bounded by y=0 and y=f(x) and x=a and x=b. The multivariable integral ∫∫S 1 dA then simplifies to the standard ∫f(x)dx. Alternatively, you could think of it as the mass of a line from x=a and x=b with quantity f(x) at each point. This quantity can be visualized as an upwards rod of uniform density, perpendicular from the line, which produces ∫f(x)dx and the "area under the curve". In this sense, the "area under the curve" is really a consequence of the more general case of adding up many quantities over a region. In this sense, the idea of adding up many quantities after some function and the area approach is really the same concept. I should say that the integral can be approximated by the riemann sum, and my textbook on multivariable calculus uses the (limit of the) riemann sum to rigorously define the integral. But the riemann sum is not the only way to do integration; the lebesgue integral and measure theory suggests that "area" is fundamental to the idea of integration.

​@@phlaxyrthanks for the detailed comment. I'm trying so hard right now to really internalize these concepts as I stare down 2 solid years of mechanical engineering courses. Every new perspective helps!

Yep, thats just the fundamentql calculus theorem, that makes the riemman iinteger summation into a continuos concept

As a physicist, I agree. In fact, if you have learned to think of it as adding up tiny pieces, you automatically know how to calculate an area under a curve. You just write an expression for an arbitrary piece of the area and then integrate over the region of interest.

It's interesting how real world application helps deepen our conceptual understanding!

@@MathTheWorld Indeed, it was rather the demonstration of slicing up balls and stuff into thin slices to find out their charge or electric field that made it click

I was about to say that, I need this idea of summing up tiny bits all the time in my statistical physics course. Thinking in terms of area in that context makes no sense

@@MathTheWorld Um, that's how calculus was invented in the first place!

That could be because having units on values makes the summing process make sense, e.g. delta-x (meters) = v (meters/second) times delta-t (seconds), so add 'em up to get a real total change in distance. Which on a graph of v vs. t, can be visualized as... the area under the curve. Later on, line integrals (such as the basketball path length shown here) are less easy to visualize as areas, but the "summing" part remains.

That’s what you get when you take the first couple terms of a Taylor series.

College teachers should definitely be placing emphasis on this perspective, but I have a feeling that at the high school level, the teachers might lack the insight themselves. Or they’re just doing the minimum they can to stick to teaching what the AP exam will cover.

I’m always hesitant with this line of thinking just because I know there is different sizes of infinity and it’s never been super clear to me when you have one kind of infinite vs another and what the implications are on various kinds of integrals. I think with single output functions mapped on a line line cut up this way you are safe though. Start combining multiple integrations together though and things might start to fall apart.

​@@Halopend You needn't worry much about combining multiple integrations together. As they are really just dividing up space along a few axes, they are still rectangles, and the terminology is simply to designate the incremental length on each axis and the range, which simply specifies how many rectangles there will be. You also needn't be concerned with infinitesimals. We are dividing a length, area, or volume by an increasingly large number, and then summing up that same number of objects again, which comes out to 1. The differing values of the successively better approximations comes from better approximations of the function being interpolated on the intervals. With the different infinities, considering the natural numbers as one infinity, the rational numbers as another, and the reals as a third, you needn't worry about those either. The main point is that we are taking a fixed distance and dividing it into a number of segments, perhaps dividing by N, or by 2^n, or by a rational or real number. The key concept is however the grid of points is laid down, each point is of zero length, width, or volume. There is no length gained or lost by laying down 10 points or 10 million. Dividing it up so and adding it up again always comes back to exactly the length of original line segment of interest; same for area or volume.

Agreed!! It has served me every day since, 52 years ago, in work and life, and thinking. I learned the subdivide and add it up approach the summer of my sophomore year, before I learned calculus. It was the hardest thing to learn--took me all summer! But everything was easy after that. I could not do the engineering calculus course when I got to college two years later, which was based on memorization rather than the foundations of math. I did abysmally. Taking the advanced calculus course, which did it "the hard way" with infinitesimals, limits, summations and approximations, and proofs. THAT I could understand, and questions resolved. Then I did excellently from then on. These concepts should be where math teaching starts. Kids can understand these--it just takes hands on and practice, practice, practice. But it serves critical thinking for a lifetime.

We're glad you see this type of thinking so useful!

That's why it's important to yalk about Riemann sums

People will still never use calculus as such.

@@w花b We had to prepare a subject and I had the normal law, I explained the link between the binomial and the normal distribution. The teacher was quite happy about it, but the other student shat their pants 😂😂 The teacher had to reassure them that it was not something to study, just to be aware since we use z-tables.

It's because teachers try to teach theory before students learn the facts. Theory is higher in most models of learning and memorization is. One of the reasons why I wound up getting so very good at math was that I don't try to understand any of it before I know what it is that I'm trying to understand. Nobody understands calculus until they know what the techniques are and the steps to perform them. Once you have that, then, and only then, can you make any headway in understanding what you know or really gain much from your experience. Trying to understand anything prematurely just ends badly.

Hahahaha

On you, not in you!

I just love when the jokes are in me

​@@haroldwiser2641he didn't learn that as well...

Didn't learn English either

Fr just use metric like normal ppl

Wow that is surprising! I definitely think of dy/dx as a fraction. It's a ratio of infinitesimally small changes in quantities. I dare say if a professor says it's not a fraction, they're wrong 😬 but any other mathematicians can fight me on that I'm willing to listen. Maybe there are certain instances where it is more or less productive to view it as a fraction? Edit: I think ratio is a better term. Fraction would indicate there is a "whole" we are dividing into. But a ratio is the comparison of two quantities which is more along the lines of what dy/dx represents.

@@MathTheWorld Well, usually a "fraction" means that two numbers are divided. Differentials like dx and dy obviously are not numbers. What "infinitesimally small" actually means is not even rigorously defined in usual math (there is nonstandard analysis, but that's not taught in most courses). So usually dy/dx is thought of a _limit_ fractions, not a fraction itself.

@@MathTheWorldMany mathematician say that dy/dx is not a fraction, however I have managed to rigorously made it a fraction. The manipulations are a bit limited however it is now a fraction and it works for multi variable functions. To do this I just introduce a value ε such that ε*ε=0 but ε≠0, and then I do df=(f(x+ε)-f(x)), so dy/dx is just a ratio of these differentials. I do have to introduce non-associativity so that there exists some values a,b,c such that a(bc)≠(ab)c, to allow you to divide by dx (which equals (x+ε)-x=ε).

@@MathTheWorld by the definition it is the limit of a fraction, which means you can't like, multiply by dx, generally I say generally because the mean value theorem frequently makes you able to make derivatives in some context a literal fraction and vice-versa see cummings' rigorous proof of the fundamental theorem of calculus for an example

Just from simple algebra within the context of linear equations based on their slopes as seen in the slope-intercept form of a line given by y = mx+b we know that the slope m is defined as rise/run which can be calculated by any two given points on that line by (y2-y1)/(x2-x1). This slope can also be defined as the rate of vertical displacement with respect to or in contrast to the rate of change in horizontal displacement. This can simply be rewritten as dy/dx. If we also consider the slope of a given line (dy/dx) with respect to the line generated by y = mx+b, the +x-axis and the angle theta between them we can also rewrite the line equation as y = (sin(t)/cos(t))*x + b which can be simplified to y = x*tan(t) + b. This relationship between algebraic linear equations and the trigonometric functions is one of the main reasons why we are able to do calculus in the first place. Without this relationship we wouldn't be able to define the secant, nor tangent function that is necessary to define what a derivative is. dy/dx is most certainly a fraction, but it's more than just a fraction. It is also a ratio, or a percentage of... for all tense and purposes dy/dx is basically the same as tan(t), it's just that they are different representations of the same thing within the confines of different contexts. Simply put, dy/dx is the ratio of the rate of vertical change in contrast to the rate of horizontal change where tan(t) is relative to the angle between the line with a given slope and the +x-axis. I would disagree and argue with any math teacher who claim that dy/dx is not a fraction. My argument becomes even more apparent when one gets into begins to work with the concepts of vectors and how much one projects onto the other. It's basically the same thing. It all comes down to some form of translation or transformation. It doesn't matter if one is asking how much displacement has occurred, or what the rate of change in something is... Without there being any logical truth to this argument then it would be infeasible to even begin to use mathematics to perform any and all physics calculations or analysis. They go hand in hand with each other. The moment that one adds one to another value two things have occurred. First is that an operator is being applied to some set of operands and the second is that a translation or displacement (movement) has occurred. One cannot do mathematics without applying physics and one cannot do physics without the ability to perform calculations. This is true for any and all fields within mathematics, physics, and other sciences including chemistry, biology, geology, geography, astronomy, business, economics, banking, etc... dy is the change in height or the sin(t) component of tan(t) of the slope equation dx is the change in width or the cos(t) component of tan(t) of the slope equation Their ratio, dy/dx is tan(t) is the slope or gradient of the line. Everything within mathematics is related. Everything we know of from basic algebra (linear equations) to polynomials to points, lines, angles, geometry to trigonometry to calculus can all be extrapolated from the expression y = x. Equality or identity within itself generates a linear equation that has a slope of 1 a y-intercept of 0, and it has an angle of 45 degrees or PI/4 radians between itself and the +x-axis. We can see this from y = (sin(45)/cos(45))*1 + 0 = tan(45) = 1. More than just this we also know that the x-axis and the y-axis are orthogonal or perpendicular to each other at a 90 degree or PI/2 radians angle. We also know that the line y = x bisects the x and y axes in the first and third quadrant which also demonstrates properties of reflection and symmetry. These properties are some of the reasons why more advanced concepts such as taking the Fourier Series and converting it into Discrete Fourier Transforms are possible. Then to be able to take DFTs and convert them into FFTs (Fast Fourier Transforms) for our computers to quickly perform analysis on various signals and waveforms is very profound. FFTs are one of my favorite algorithms. All of this is possible because of the relationship between linear equations and the trigonometric functions based on their transformations from translations to rotations. I'd like to see these math teachers build a working 3D Graphics/Game Engine from scratch using either OpenGL, DirectX, or Vulkan in C/C++ with a full working physics/animation engine, and then claim that dy/dx is not a fraction. Good luck with that. Good luck with performing Matrix and Vector Calculations and applying various transforms! Or I'd like to see them build a working 8-Bit CPU from basic circuitry... Yeah, that's not going to work to well in the real world if one is under the assumption that dy/dx is not a fraction, ratio, or percentage. Yeah good luck with that one.

Thank you! And thank you for sharing your example of how this type of thinking has worked in your field of study

That's great! Lots of other viewers have mentioned that too with their physics classes

@@MathTheWorld I also had a ChemEng teacher, on a math modeling class, that had an equation written in the blackboard and, at the start of the class, asked us how much is infinty. After a lot of mathematical answers, he said it was 5. Then he substituted 5 into the equation and the transient part was less than 0.01% of the rest, thus we were on steady state. Then he said "welcome to engineering".

@@ciroguerra-lara6747 "Mr. Owl, how many licks does it take to get to the center of a Tootsee roll pop?" "Let's find out. One ... twoo ... threee ... crunch! Three."

@@MathTheWorld Maybe PhysicsTheWorld is the way to go instead~

Especially Gauss Law in Physics. It really made me confused for about 7 months for having a bad teacher

The teacher in the statistics/probability oriented maths class was also rather boring (though; I hadn't got my ADHD diagnosis yet, so that applied to lots of teachers) but at least she was curious enough to actually bother to go on a tangent* when we asked a question even though it was outside the current learning plan. And though I still find statistics and probability to be the less fun part of maths; she was the first maths teacher I had (when I was 17-18) who actually taught me that there could be anything fun and interesting to learn in maths even when it wasn't applied to any real world problem. Every maths teacher before that, just taught me that maths was some sort of horrible punishment of mindless repetitive writing and symbol juggling that we were forced to endure for some unknown reason except the promise that we would need this next year for more mindless mental torture. I'm sure if RU-vid had existed at the time; and I could have learned maths directly from professional and recreational mathematicians online who actually enjoy the intellectual game that maths _really_ is; I would have been so much better at maths than I am. Because then I wouldn't have wasted 10+ years being taught that maths is terminally boring. (* the metaphorical tangent not the geometric one)

To be fair; even the crappy maths teacher actually told us that both the Σ and ∫ symbol both came from the letter S in greek and latin script respectively. Though only my physics teacher had an explanation of the difference: basically that it's just a convention that we use Σ when we're doing actual discrete sums and ∫ when we're doing the infinite sums of infinitesimals or limits however you prefer to think about it. While the idiot maths teacher thought infinitesmals was somehow "heretical" because limits was the "correct" interpretation; because whoever it was who introduced the idea of limits had "won". At the time I had no problem understanding the infinitesimals idea and really struggled to understand limits because to me it seemed either like some sort of handwavy mental trick or just a very obtuse way to say infinite sum of infinitesimals while using all sorts of wishy washy terms just to avoid the idea of infinitesimals. Today I understand both approaches; but to me they still seems like just different ways to think about the same thing; so the only thing I still don't understand is why so many mathematicians still seem so uncomfortable with the whole idea of infinitesimals, and why so many even seem to think that "infinitesimals" is "wrong" somehow. I get that it's hard to imagine that an infinity of zero width (or rather a width that is smaller than any number but greater than zero) slices could add up to a number; but you have the exact same problem with the limit thing except here you go smaller and smaller and then you close your eyes and put fingers in your ears and ignore the zeroes and infinities and chant "It APPROACHES infinity, it never gets to infinity, it APPROACHES. The LIMIT, the LIMIT. Theres no such thing as an infinitesimal!" or something like that 😉 The "problem" of infinitesimals vs limits is purely theoretical anyway; just like infinities are also purely a mathematical theoretical construction anyway. Reality doesn't appear to be infinitely divisible anyway nor contain any real infinities (at least not the observable part anyway) so in real life a billionth or a trillionth slice is more precision than you'd ever need anyway. If the infinitesimal idea makes you uncomfortable, think of it as not an infinity of infinitesimals but a really really large number of Planck lengths; that would be just as accurate and also avoids the zero times infinity problem too.

your feelings are irrational

⁠@@SteinGauslaaStrindhaugthe reason mathematicians don’t like infinitesimals but do like limits is because it is much easier to define limits rigorously, this does not mean they are easier to understand for beginners. there are however ways to make infinitesimals rigorous, it’s just that old habits die hard.

@@Fire_Axus Nah... I'm pretty sure my feelings are integers, surely they must originate from a finite and whole number of electrons, molecules, and ganglia. (Maybe they could be rational, if they come from the ratio between those things.)

>"Basically you are using Riemann Sums within the context of discrete step intervals where the step intervals have a defined width as opposed to the pure integration" First, the limit of Riemann Sums IS integration. There is no difference the two when integrate over an interval (measure theory is irrelevant to topic at hand). Riemann Sums is about limit... about infitisimals... it is pure integration. If you really want to make a distinction, then limit of Riemann Sums is more important and general than integration of real valued function at a symbolic/algebratic level. Students wrongly think the later to be more important. Second, the point is student should understand integral as limit of riemann sum (or generally just summing up small pieces... like finding a finite time evolution of a generic mathematical object like a time-dependent temperature field, or time-dependent vector, or time-dependent density matrix... by summing up small parts of the mathematical objects, ie the infitisimal time evolution). This understanding of summing up known small step contribution to find unknown large step contribution is the essence of integration which most student don't intuitively understand. The "area under curve" is useful to first learn the idea of Riemann Sums and integration as limit of Riemann Sums. However, most student stop there and only associate integration with area under curve while missing the more fundamental and useful idea of summing infitisimal pieces

@@bohanxu6125 First, I'm not a student. I graduated a long time ago. Secondly, you missed the entire point of what I stated between the two. One is within the context of that which is finite, where the other is within the context that is purely infinitesimal. The Riemann Sum is not exactly the same thing as an Integration over some bounds. It only converges to one when we decrease the size of the width of its step interval and the iterative index increases or tends towards infinity. It's only in this case at the point of convergence that they become equivalent. If we take a curve of some function f(x) and we set a bounds at f(a) and f(b). Yes through integration there is a definite area between [a, b]. This is a standard approach at evaluating a definite integral. If we break this region down into smaller subsections where these subsections have an area defined as (f(x+dx) - f(x)) * dx where dx has some defined width that is not 0 then the summation of these smaller subsections or partitions of that region approximate to its true area with a small margin of error epsilon. They are not exactly the same thing. Yes they are similar and only when certain criteria is met is when they then can be interchangeable. Being similar doesn't make it the same thing. It is only when we decrease dx to 0 and the amount of smaller regions increases towards infinity is when these two are considered equivalent. This is the application of applying or taking its limit. It's only at this point that the Algebraic Form of Summation becomes an actual Integration. You might want to refresh yourself on the Fundamental Theorem of Calculus.

@@skilz8098 I was trying to say "limit of Riemann sums IS integration" (but being a bit careless at times), but yes Riemann sums technically means finite approximation in most textbook convention. Still this is essentially the problem. Student think integral (or area under curve) is more "fundamental" than Riemann sums which is "just an approximation". This is a failture of teaching. In reality, limit of Riemann sum IS the integral. This summation of small known contribution converging to large unknown contribution, is the essence of integration and calculus in general. Understanding Riemann sum and limit, is way more robust/fundamental and useful than understanding integration as area under curve or symbolic maniputation of analytical function. This is my point, and point of the video. Riemann sums is not just a less important approximation of integral. Its limit is integration... and student should only use area under curve as a special case to learn the more general and useful idea of riemann sum (and its limit). Again, if one simply understand integration as area under vurve or symbolic manipulation, one essentially can't undertand concept like divergence in vector calculus. Those things only make sense if you understand calculus as summation of small pieces (the limit of).

​@@bohanxu6125 Yeah, that's the key part between the two. It's the limit itself and when we apply the limit of a Riemann Sum that it begins to converge into an actual Definite Integral. That is the fine distinction between the two. A Riemann Sum without taking or applying the limit is still Algebraic. Applying the Limit is the Calculus of it. Yet for most tense and purposes a Riemann Sum and a Definite Integral in most cases are interchangeable. They are very similar although they are not exactly the same. This is also why they have different notations. And the reason I state this is because if there exists some Riemann Sum that doesn't converge but instead diverges then that Riemann Sum is not equivalent to a well defined bounded Definite Integral. I think this is something that many people tend to overlook. It's only when a Riemann Sum converges as dx decreases that it becomes equivalent and interchangeable with its Definite Integral counterpart. There has to be a convergence in order for the two to be considered equivalent. Now, does this suggest that there may exist some Riemann Sum that doesn't converge and diverges instead? No, not necessarily and that would require a rigorous proof to solve. This assessment of mine is more of a conjecture as it's based on the logic that if the process of summation of smaller parts converges towards the actual real value then the equivalence of the two is true, otherwise if it diverges then it is false. And for the false case, the summation and the integration are not the same.

@@skilz8098 I don't want to sound mean or anything... but I don't know if you get my point or not. I think it would be productive to explicitly acknowledge our agreement with each other. Student shouldn't think integral just as area under the curve symbolic manipulation of functions, because integration from its raw definition is fundamentally a numerical process about limit... and because this numerical understand of calculus is more useful and general (because it is the raw definition). There are case (like divergence in vector calculus) where integration doesn't associate with area under the curve, nor can be solved with simple analytical function. One must think integration numerically (and as limit of numerical value). The fact that some problem can be understood as area under curve, or symbolic manipulation... is just sheer accident. Also if we really want to get technical, I'm decently sure convergence of riemann sum (when the right definition is used) is totally equivalent to well-defineness of integration for intervals with reasonable functions (again, measure theory about pathalogical function or region of integration, is irrelevant to topic at hand. also even measure theory is just riemann-sum-like process with a more robust interval choice). When integrating over intervals or non-pathalogical regions in R^n space, integration is exactly the limit of riemann sum (given appropriate definition of riemann sum). "Riemann Sum that doesn't converge but instead diverges then that Riemann Sum is not equivalent to a well defined bounded Definite Integral." simply doesn't happen, because limit of riemann sum is integration (for integration over non-pathological region in R^n space, and with reasonable deifnition of riemann sum). Thinking Riemann sum as less robust/fundamental or "just an approximation" to integral that is symbolic manipulation or area under curve, is precisely the failure of teaching. One can still talk about integration in pathalogical cases through measure theory. However, student should first understand integration as limit of riemann sum in non-pathological cases because this is the concept that is actually useful in real life where the pathalogical cases almost never happen.

We understand. The scale we had on hand was in pounds 🤷🏼‍♀️

yeah, it can cause some deaths

Thank you 😭

I am so glad it was helpful. Thank you for the positive comment!

Why?

Poincaré was a great mathematician

It is in the works! Give us a few weeks.

I am finding AI very helpful in getting all kinds of questions answered, physics, math electronics, history, microscopy, biology--any old thing that comes to my mind. I use Edge browser with Copilot version of ChatGPT, and ChatGPT on my phone. It's like having an assistant to read hundreds of sources and come up with an answer. I already know a lot of the underlying principles, so I can quality check the answers, and careful wording can redirect the AI to more meaningful or valid answers. Similar deep dives using google search in the past would take me days of endless searching. Now I can do the same in a few hours, and build a spreadsheet to do estimates and computations.

We're glad you found it so insightful!

In this video, google sheets found the polynomial. You can make google sheets find an approximation for any data set relatively easily.

This is correct we just used google sheets best fit line feature and asked it to do a 4th degree polynomial! But regression lines would be an interesting topic!

​​@@MathTheWorldhaha yeah. They are handy. But I have too often seen children in highschool being taught regression functions by just being told to enter a table of numbers into desmos and write down whatever function it spat back out as their answer. No least squares. no gaussian elimination. It felt like a data entry job disguised as a math lesson. So many techniques of doing things by hand have been unfortunately lost over the years. I'm pretty sure I'm one of the few people in my age group that knows how to calculate Common Logarithms and Square Roots by hand

To understand regression rigorously you first require calculus.

Thank you!

I don't get the "cleaning up" part.

@@__christopher__ In "Electric" integrals there are many things not relevant to that integration. You can pull them out of the integral and usually you are left with something rather simple to integrate.

Glad it was helpful!

Yes surface area and arc length are two big ones!

That's the plan!

Energy**

Great job coming up with it yourself!

Exactly! That's what makes our basketball example so pertinent. Trying to find area under the curve doesn't make sense there. You're finding the length of the arc path traveled by taking the wide route and comparing it to your defensive player's path

I did not see that example reasoned out. You only showed the first one, reinforcing "area under the curve"

Yeah, but that makes sense as those that study integration itself aren’t physicists.

That's a great idea! We'll keep that in our back pocket thank you

The commonly used integration symbol is itself a stylized S. It's essentially S ____ dx or whatever else. It's basically just an evolution of the previous ∑ multiplied by the width of each of the elements. If the width approaches zero, and the number of elements approaches infinity, you get an integral out of it.

They are very connected! At least adding up tiny bits of area helps understand the area under the curve conception really well. We use it all the tiny in understanding how volumes of revolution work! The best example we have, and plan to do a video for, is Arc Length! That's one type of problem where the area under the curve conception does not work very well but solely adding up tiny bits of quantities does!