What is Integration? 3 Ways to Interpret Integrals
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- Опубліковано 12 лют 2024
- Integrals Explained! This video explains 3 ways to understand and interpret integrals in calculus. Two of these ways are commonly taught in school. But one way is often left behind but really unlocks real world problem solving! These conceptions of integration are then applied to real world problems.
Link to research article on student conceptualizations of integration: www.tandfonline.com/doi/abs/1...
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The claim of this video was not clickbait! That absolutely helped me understand it better and it should be the default way of teaching this subject.
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.
As a senior in mechanical engineering, I can attest to the importance of this. When you’re looking at heat flow or force concentration, a lot of the methods and equations are derived from this idea of adding up many tiny bits after applying some function to them. Good work here!
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.
Jokes in you. I didn't learn it at all
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
I really understood integration after taking Physics, it helped me think better of it as adding up tiny amounts of stuff
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.
Just so you know, at 1:40, bows bend when you pull the string back. You aren't stretching the string. Also, the reason the force curve flattens and gets lower at the top is because of the additional mechanical parts of a compound bow that are designed to make it easier to hold at a full draw.
I cannot emphasize enough how much the idea at 3:35 has been useful to me so far.
The fact that any (usual) function looks like a line if you zoom in enough opens so many doors in limits, numerical approximations & computing, not to mention linear algebra.
It's crazy how teachers gloss over this
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.
This is absolutely how I think of integration. Whenever people try to say they'll never use calculus, or that it's too complex, I always tell them that calculus is the math of rates and totals, which, I think, are both very fundamental concepts for interacting with the world.
And they're both very nicely captured, as you showed, by discrete problems, which are much simpler to conceptualize. The derivative corresponds to the difference between consecutive elements of a sequence, and the integral corresponds to the sum of consecutive elements of a sequence. By expanding out the terms of a sum of differences, you find that tons of things cancel, yielding the analog for the fundamental theorem of calculus. They even have a close analog to the product rule, and therefore, integration/summation by parts. I think it's often a missed opportunity in math classes that they don't clearly make these connections. Great video!
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.
@@whannabi 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.
Great vid, but doing physics using feet and lbs should be illegal
We understand. The scale we had on hand was in pounds 🤷🏼♀️
yeah, it can cause some deaths
I always enjoy the moment I tell people about the integral symbol just being an elongated "S" because it's a sum, and just watching the light bulb turn on in real time. Very satisfying
I definitely found this was one of the biggest differences between calculus in maths class vs later on in engineering.
With questions like "is dy/dx a fraction?", a maths teacher would say no and (understandably) focus more on rigour and strict definitions. Meanwhile, engineering professors would be much more cowboyish and would freely treat dy and dx as any other variable. Obviously mathematical rigour is necessary, but sometimes focusing too much on it can obfuscate the core ideas and distance people from a more intuitive approach
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.
In my school we were asked to draw different standard shapes with provided dimensions like a square, triangle, circle, hexagon etc on a graph paper. Then were asked to add up the tiny 1mm squares to find the area. We were then taught the formula and we were also explained how that formula can be derived using what we were doing on the graph paper: adding and multiplying many small things. Later we did that to volumes as well and finally 3D vectors in graduation level math, electromagnetics etc. It is surprising to me how so many comments are saying they didn't do this or their teacher glossed over this.
no we didnt add up squares to find the area. we just memorize the formula of area without understanding the concept. thats not suprising to many asian students.😂😂
An old astronomer told me they would figure out the area of a galaxy or nebula by cutting it out of a photo printed on card-stock, and _weighing_ it! Saves counting all those squares and partial squares, You can do the same thing with an elliptical orbit to verify Kepler's 2nd Law (equal areas in equal times),
must have a great school
@@siammahmud-jb1ex That's the thing it wasn't a good school like fancy or top tier. I was with my maternal grandparents as a child (age 5) across the border. When my parents decided to bring me back to their resident country, none of the good schools wanted me. So I was admitted in whatever school took me... I would call it decent but still below average. The teachers were definitely mostly old and not the young teachers like today. Hence maybe their only passion in life at that age was educating the country's future? That's my guess.
Bro... my school barely taught Algebra 2
I always felt that to understand a tool or a solution, it was very important to understand the original problem and the initial solution that gave rise to the refined solution.
If you watch this video in reverse you will learn differentiation!
My physical chemistry teacher, with regards to classic thermodynamics and steady state transport phenomena, used the concept of a "physical integral". It is basically what you described.
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~
I was just wondering why my modern physics lab had us work out the sum of a bunch of discrete averages instead of just integrating some given wavefunction, now I get what they were trying to do. Great video.
Thank you! And thank you for sharing your example of how this type of thinking has worked in your field of study
I wish I could watch your explanation 18 years ago. This is so delightful to see.
Can vouch for this. Really struggled when I moved on to physics in college, wasn't until years after that I realized this was the issue.
This is a major reason I feel like I understood and learned so much more maths in the physics class than in the maths class... Derivation and integration is so much easier in physics; it means something: finding velocity from distance, distance from velocity, acceleration from velocity, and jerk, snap, crackle, pop if you go on. And you "calculate" the integral simply by summing up the slices under the curve, and find the derivative by examining the slope at every point; and since it's physics and the curve is not a mathematical construction usually but a measurement in the first place you don't have to fiddle with the sumbolic derivation/integration since there is no actual alternative than to estimate/interpolate the values from the curve/plot itself.
Also my maths teacher was terrible, and had absolutely no comprehension of how anyone could not understand maths in the same way as him immediately; so I eventually switched from the physics/engineering oriented maths class to the statistics oriented class to get another teacher.
I think I do remember the symbolic way to do it; but it was _sooooo_ boring the way it was done in maths class because it was completely divorced from any practical application, and because the maths teacher was an uninspired idiot; he didn't even connect it with anything interesting within maths either. It was purely mindless busywork with no insights or application or fun at all with him.
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 UA-cam 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. The main difference between the two is that a Pure Definite Integral is the true area under a given curve where the Riemann Sum is breaking it into even partitions and is a very close approximation with a small margin of error denoted by epsilon. Other than that, they are comparable and are nearly interchangeable. We can easily convert or substitute most Definite Integrals into an equivalent Riemann Sum and we can convert most Riemann Sums into a Definite Integral based on their definitions through the fundamental theorem of Calculus.
>"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.
The tiny-bit sum conception is especially useful when you start working with surface and volume integrals.
Especially Gauss Law in Physics. It really made me confused for about 7 months for having a bad teacher
I loved the fact that in high school I dual enrolled in physics and calculus. And then math finally made sense (I'd been taught mostly memorization) and on the advice of my calculus teacher, went back to 1st grade my senior year to learn how to subtract using my fingers.
Integration = The sum of many multiplications 🤣
This is 100% true, and something I like to tell all of my friends that are struggling in calculus. When I was in AP Physics 1, I had heard that moment of inertia formulas could be calculated using integrals, and because I had no idea that an integral was anything more than the area under a curve, I couldn't conceptualize this. Eventually, I went on a long derivation trying to find the formula of the moment of inertia of a cylinder using the summation form of an integral (limit as the upper bound approaches infinity), and I finally made the connection.
Thanks for the great video! I recalled my time in high school and remembered how my AP Calculus teacher put in the extra mile and made sure that students not only view integrals as simply the "area under the curve" or "anti-derivative of a function." Instead, he solidified the idea of summing up small bits or small sections from a particular function, comparing the results obtained by integration and manual summation by means of Riemann sum. Needless to say, this foundation really helped me a lot during my undergraduate studies in Material Science and Engineering, with one memorable moment where I used summation added up tiny bits of some tabulated data during my Thermodynamics of Materials finals to calculate the Entropy and Gibbs Free Energy of Mixing of metal alloys!
Oh man a video on why we can use an antiderivative to calculate an integral would be awesome. My college professors never taught me it and I’ve always wondered how that works. Great video!
It is in the works! Give us a few weeks.
I just intuitively understood why defining an inner product with integrals makes sense in a broader way than just checking if it satisfies the properties. Thank you!
The first time I came across integration in the sense that you are talking about is in a physical modelling software library that integrated force over time to find distance
I had this problem in high school and finally learned about the limit of sums definition in college, and that helped a lot.
I still struggled, but it helped once I figured that out.
Great video!
An absolute must for first year physics and engeenering!
I wish I saw this on Sunday, could have helped me with my final exam
As someone who essentially taught myself calc 2 (took it asynchronously online) this short video gave me more of an understanding of integration than my whole course did 😅
This makes cody from cody's lab a legend. He simply plot the graph, cut the paper and weigh it to find ratio of the area under the graph.
I’ve taken Calculus II In university exactly 3 times. This video is the best explanation of what calculus actually is, that I’ve ever seen.
Thank you 😭
It has been probably 20 years since my last math class, calc in 3 variables. I remember distinctly learning this property of integrals in my pre-calculus course and then we really didn't deal with it much. It was used as a tool to illustrate the concept, but then, we never really progressed with it. As you point out, this method is tremendously useful because it allows us to apply the mathematics to many situations in which we wouldn't naturally think "hey, it's calculus time!" For me, that's probably a good thing anyway. I've forgotten so much math as I got into a memorization-heavy field (medicine) that I probably would have to retake a course to do it. Rambling aside, thank you for the video explainer and accessible refresher!
A video must-watch for every college student.
I gained this intuition when working with data, i.e., discrete math, which is your key point in this video. Once you introduce the concept of differential 'dx', you travel to the domain of the abstract continuum; I say abstract because 'dx' is just a non-measurable concept.
This is genuinely one of the best integration videos I've seen and explains probably the most useful thought process of coming up with solutions to real world problems with changing data that I have ever seen!!! I also think that thinking of integration like this makes calculus so much more fun (beyond just being the underlying way integrals work), it gives students a more intuitive way to approach problems that almost anyone who sits with it for a while can easily do.
We're glad you found it so insightful!
"Adding up tiny bits" seemed like the most realistic interpretation, infinitesimals notwithstanding.
When I learned integration, We would also talk about Integration in our physics class. To show where exactly integration is used, and what it is used for.
For instance when we used the equations for motions with linear acceleration (finding distance, or start speed, or acceleration) Those formulas
are derived from integrals or derivatives.
We also looked at a lot of practical examples like, finding the volume of objects using integration, particularly cylinders, where the radius along the length of the cylinder is expressed as a function. Thus we find the volume by using the integrals of the function.
Finally! Someone actually explains what integration actually is. I’ve been trying to explain this and its importance in solving actual problems for so long
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.
Interesting as I had three very good calculus professors who had also written a very good book on the subject. Larson and Hostetler were the main authors, but Heyd did the solutions manual if memory serves. We were taught mainly that integration was adding up tiny parts and a key aspect of integration was how you set up the tiny parts in relation to the shape to be integrated. That served me well in my engineering career.
When I learned integration we covered Riemann sums so I think this idea was always clear to me. Though perhaps more so when studying measure and integration theory and then thinking of other applications of integration. For example it occurred to me that if you have a topological group with a Haar measure then you can define a group ring structure using a convolution to define the product.
My college math teacher made us work summation of Logarithms for two months prior to actually teaching us integration to pound home the point that integration is the new way of solving old problems. When he finally taught us integration I was initially mad at him for wasting two months on summations. However when I got out into the real world I was surprised how many engineers did not understand this.
Honestly, I was taught from the beginning to treat integration as sum of small pieces. You are right, it so easy to see where integration can be used from that perspective and how to set up problems. This perspective teaches how integrals work. The other two just show specific applications without teaching how to use the tool or why it works. I always hated being given a formula to memorize without being told how it was derived. How do I know how and when to use a formula if I dont understand how it was derived, what assumptions and approximations were made, and what conditions apply to it.
I believe also thinking about derivatives & integrals as the rate/anti-rate of change in quantities is important because you can apply that thinking to any real world situation that changes over time. Derivatives give you the “specific value” of a large quantity & integrals give you the “large value” of a specific quantity.
(Just started doing calculus sorry if I'm wrong ig 😅)
So basically it's the opposite of differentiation, this is where you break the curve into teeny tiny pieces and find the rate of change. Integration is the opposite so you get thr tiny pieces and add them up. And because they are. infinitesimally small the rectangles below (area) get closer to the actual value the smaller you go as the closer you look the flatter the longer and the smaller the little areas between the line and rectangle is and so this value is fairly accurate.
You have a very good approach to these applied problems .
Thank you!
Perfect! Thank you very much for this simple and lucid example, perfectly explained. 🙂🙏
Glad it was helpful!
Very cool video, but at 6:56 what does the R squared mean? And how did you get that function from your measures?
I hope u continue this series. The start is great. Thx
That's the plan!
Loved this video! Thank you.
sometime ago i really thought hard about integration, beceause it frustrated me that i have a good understanding of derivatives but not Integrals. The conclusion was to forget everything that has to do with "area", because it is only a byproduct of the interpretation. Further i came to the conclusion that all i do is just multiply a ceartain change (derivative) with a really tiny interval (dx), which not only means the tangentline of my antiderviative but also that i sum up all these small changes (which can obviously vary) with the same dx. So i get something like a state. It is truthfully rather hard to explain it in words... It's more akin to an epiphany. But this video is a great help for those that need the intuition!
What would example integral equation be for the solar panel and basketball question be? I feel like I dont fully understand it yet, and having more than 1 example really helps
Also when the average force for each interval was calculated, was it an important step that you chose the average of 2 intervals, or was that just an arbitrary step that you could take if you want?
When teaching about integration, I've often found that doing discrete calculus (using the discrete derivative, aka finite difference, u'(n) = u(n+1)-u(n), and the discrete integral, ie, SIGMA u'(n) ) have been tremendously useful in getting this vision of "adding up thin rectangles" across. In fact, you can also get the intuition for the fundamental theorem of calculus for free, since it's just a continuous analogue of telescopic sums.
this is a fantastic video, and I'd love to see you show how the antiderivative works
Amazing!! somebody finally said this!! A few months ago, I happened to solve a complex problem using 3rd kinda thinking. Hoping to publish the research soon
It's been a while since I've taken calculus, this is a nice intuitive refresher into the idea of integration.
I was lucky to have been taught Integration first in physics, we derived all kinds of results like Electric field due to a disc and moment of inertia of a spherical solid, with varying densities etc. We learned the concept of Integration as adding up contributions of very tiny figures. It was only later that we were taught In depth about Integrals in mathematics. Where we learned about different kinds of substitions, Error functions, Reimanian sums, etc.
This is the intuitive way to calculate. You look at something curved and to get a good estimate at how much material you need to "fill" the curved area, you split it up into easily measurable sections, and add them together.
It was such a mindblow for me when I figured out the abc formula was so important because we often approximate things to be second order equations (like the spring force)
Energy**
I had a physics test in which there was question where we had to calculate the velocity of a body that falls from a height h (h is large) from the surface of the earth. Since the acceleration varies with g=GM/R², I started using integration to solve it and i got an answer and I felt so proud for using such an ingenious method. Turns out it was the wrong answer and you just had to equate the change in kinetic and gravitational potential energy 😐
This is really key to being able to get any meaningful understanding. Area under the curve really only works for very simple cases, and immediately fails when trying to do things in higher dimensions, over surfaces, etc. Antiderivatives is sort of the punchline without the setup - it's the brilliant insight of calculus, the eureka moment that tells you how to calculate an integral, but it doesn't really tell you why you would want to take an integral in the first place. I sometimes wonder if it would be better to start the conversation with discrete sums using sigma notation, and then pivot to integrals just being a continuous version of that.
Yes surface area and arc length are two big ones!
8:51 fundamental theorem of calculus is a special case of 5 related theorems that they are essentially the same theorem applied to different context:
1. The fundamental theorem of calculus
2. The fundamental theorem of line integrals
3. Green’s theorem
4. Stoke’s theorem
5. Divergence theorem
These 5 theorems is essentially 1 theorem applied in 5 different cases.
Yep, did all three of them in 1991, starting with option 3.
love this! Would also love to see more videos like this with real world examples
Thank you! Our channel is dedicated to showing how math is used in real life. So definitely more to come!
Summation was how we first learned integration and that led naturally to reversing differentiation. What was missing was seeing integration as the creation of a new function which gave the area.
Very useful for multivariable calculus - a common course taken for first year undergrad students.
Damn if ppl aren't thinking of integration as adding tiny bits of a shape then they are missing out on the beauty and usefulness of integration. How it made some very hard prblms possible and almost intuitive (after you have practiced it a bunch).Glad to have teachers who engraved that in my head by saying it again and again.
I would say this is is more akin to thinking of things in terms of functionals rather than integrals which are a subset of functionals.
Nice way of explaning, especially the practical examples.
To me these 3 Things always were representations of an integration, but what it is is the calculating what the rate of change at every moment combined results in. Suppose a car drives 3km/h for 2s and then 5km/h for 3s if i add up the rate of change in every single point i get the distance traveled from start to finish. Of course i don’t necessarily know the starting point hence the +c in the end. The other ways of thinking about the integral were always just parts of the whole thing for me. Of corse this is the area under the curve and of course it is adding up tiny bits and of course its the antiderivitive, but only as a whole it clicks in my mind.
6:35 how did u build that polinomial expression fro those points? have u made ordinary least squares and then what is required to identify that function?
We actually just put the data in to google sheets and created a scatter plot graph then enabled the best fit line feature and asked for a 4th degree polynomial.
Yes, I would. My issue is thinking about how you come up with thinking about the data.
How did you know to calculate the average of each endpoint of the interval, I would have just calculated how much force it took to get from one interval to th next in 4:30
😃 Love this! A great reminder to consider how to simplify a problem _before_ embracing its complexity
Great presentation. I often regret the quality of classroom instruction i received when i first encountered this.
Great video, thought at first you were a popular UA-camr with the videos quality
Thank you, brother
As a calculus teacher at the university for 20 years I partially agree with you. The excessive focus on the antiderivative (completely useless in many applications) and area calculation can lead to two misconceptions:
One is that some integrals "can not be solved", and that's wrong because finding the antiderivative is not "solve the integral".
The second one is to think that the definite integral is the area under the curve. WRONG! It's related to the area under the curve but it's not the same.
Amazing job on explaining the "how," congrats. Would be great to see your take on "why" does calculus work at all in the same digestible manner
How do you get from 2 place accuracy on your input data to FIVE place accuracy on your results?
That 3rd definition, summing tiny bits, was my first thought for what integration is. The trick? I'm a physics major...
If you're not using this definition by the end of physics 2, you're going to start thinking about it that way in subsequent courses.
Great video! One pedantic nitpick from someone where this is their pet peeve. Further != Farther. Farther is a measure of distance, and further is a measure of degree.
This is awesome! I was just complaining to my second semester physics students yesterday about how the math curriculum generally fails to capture the meaning of integration as a summation device for adding up continuously changing infinitesimal contributions (we are doing a ton of electric field integrals right now). I have the advantage of wearing both hats as a teacher (math and physics), and I try to drive this point hard when I teach second semester calculus. z
As a math major, I feel like I use this property less, but its still definitely important. I think a big thing Ive seen in calculus classes is introducing the Riemann Sum (the law that teaches this very concept), but often don't connect it well to the concept of integration, so it feels disconnected, and often my non-math friends get confused. Wish there was a bit more care for this part of integrals in calc classes.
the fact that Leibniz thought that it was important to remind you of the sum when modeling the integral symbol tells you all you need to know.
I have done undergraduate calculus. But this is what I have been doing in real life - integration using Excel sheets.
I see two different realms, the discrete and the continuous. In the discrete realm there is summation of tiny parts. In the continuous you have integrals. The former is used to solve practical problems. The latter for mathematical or theoretical problems.
My undergrad calc courses did emphasize "adding up tiny bits", mainly with lots of problems involving computing the volumes of various solids in various ways. EE signal processing courses are the only time I can remember using integration as anti-differentiation (purely for math simplification purposes with no physical meaning). And "area under the curve" is "adding up tiny bits".
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!
"Summation of infinitely small quantities" by I.P. Natanson. Great book.
Great Video!
It show us how important to do experimental's and collecting data, because you are not always will get a function to integrate in real life problems.
In school our teacher said always "summation is integration" for us to remember. I remember we used it to calculate some potential energy, integrating over gravitational force along the Distance in physics class.
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.
this is very interesting thank you!
Glad to know I learned integration right. Thanks Mr. Andrews!
Really hope you make more calculus stuff! I always had trouble conceptualizing when i go to the 'infinitinely many infonity small bits stage' (if someone has a good way to explain this lmk.) Awesome video!
We plan to!
I know that this video is primarily about Calculus, but the one thing I kept fixating on in this video was how you came up with such a precise 4th degree polynomial to approximate the data points.
Is it possible to do a video sometime in the future exploring regression functions and a rigorous way to come up with one with a decent r²?
Perhaps you could also discuss how to calculate the r².
Unfortunately regression functions is a topic that's often neglected in schools.
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.
I think you could have used Hooke's law and some trig to get an equation relating the force applied on the bow to the distance from equilibrium. Can it work that way?
The ideas in this video are really important for setting up integrals, which is often the hard part of a real-world integration problem. But I'd argue that, without the antiderivative technique, we don't yet have the main idea of integral calculus, which is that, if you set up an integration problem in this way, you can (at least sometimes) use the antiderivative technique to solve it. Otherwise, you can formulate the problem, but it looks like it would take an infinite amount of work to solve it exactly.
Great video! Would love to see some more vids on applying calculus to the real world.
The bow example is sensless. What about if I wanted to pull on the string into infinity and then integrate from 0 to infinity?