I'm 71 and did a lot of calculus while studying for my master in engineering. Well, now, 50 years later, I don't remember anything about it. Thank you for this little insight. I enjoyed it.
I'm 72. Welcome to my world. I have two entering degrees, one in electronics and the other in electrical engineering. I was once buff, pretty (this is a joke; I'm a male, lol), and bright at least once. I have since forgotten most of this stuff. Computers came to the rescue in the 90s. High-end calculators, like those on smartphones, took care of this for me. I have a learning disability-bad memory. I'm not able to remember formulas. But nothing I've built or repaired blew up.
I took the 3 required semesters of Calculus in College for my Comp Sci degree and never used it again(tiny bit in my MBA coursework). Just here for a refresh 40 years later. I just remember it being rigorous with 2-3 hours of homework each time we met. Now that I'm retired, I get to learn just for the fun it. Onward into Python now.
I'm 76 with a Bachelors degree in mechanical engineering technology. This brings back a lot of memories of my technical calculus classes. I still go over my calculus papers and books.
I'm 69 1/2 years old and hated mathmatics since childhood. I struggled through high school and college with the bare minimum courses in algebra and geometry just to get through school. Thank-you for showing me in your kind calm way how this works. I won't become an Einstein, but at least I can go to my grave saying that I actually understood a basic look at calculus and understand why it is important to our civilization. Thank-you for that, sir!
I'm 67, and have a degree in Aeronautical Engineering. I struggled thru calculus in college back in 1975-76 and after that year, I never touched Calc again until a few minutes ago. I think I agree 100%$ with you!
@@QuantumRift I'm an idiot, but having calculus described as finding the area of weird shapes, learning that you don't use it in aerospace engineering is mind boggling to me
I studied Calculus in college, then on Engineering..scored distinction. Then worked...worked ...then worked in Engineering field.. Retired now. Still waiting for the day to use Calculus knowledge.
well I don't know. I can follow it but there's still a lot that is all Greek to me. I guess like he says if you know the rules you just apply them and you got the answer-but that's where I'm lost. I don't understand why and where you get the stuff following the integral sign. A bit of it maybe like the 2 squared and the 3 squared but that's it.
I read this and had to recheck your name, thinking, 'did I post this sometime in the past and forgot?' 65 1/2 here and exactly the same (including two hot summer remedial sessions in math). but after my recovery from high school I regained an interest and went back to night school for another go, at pre-calc and calc. Failed but completed and it rekindled something teachers had nearly killed off years ago. A good presentation. I could've used some of the gaps filled in ("so, that works for THAT shape, but what about ....") but I get who the target audience is and I'm still in that group. ; )
I'm a short distance to my 79th. but still following maths in U-tube. I have already learned matrix and now I'm glad to start Calculus with you. Never did in my O-level back in the late 60s. Thanks from Zanzibar
It's amazing how the things that sucked balls for us at high school interests us as we get older. Maths and history for me. Maybe we're aiming this whole education thing at the wrong age demographic! 😂
@@GTKJNow yeah, at that stage he just showed a rule without any explanation behind that rule - so at the end of the day, it didn't really shed that much light on calculus other than the titular concept of summing up the areas of thin rectangular strips beneath a segment on a curve
How many 50+ people are watching this video?😊 I lived half of my life as a soldier and the remaining in medical field, just curious to know if there are other people like me who missed math. 🙂🙏
I just wanted to say a Big Thank You to John, so I hope he reads this. I am just finishing his Algebra 1 course and completed his foundation course before that. At the age of 62 I decided to confront my fear of maths, it was some unfinished business since I realised why I had developed a belief that I couldn’t do it at high school and that I should be able to overcome that. For the past year I have worked hard and have really felt that John was at my side and on my side - even though he does not know me. I have understood everything and to my surprise and delighted I have found that actually, yes, I can do math and moreover I love it! I love the way he makes such an 13:37 effort to explain things clearly and methodically. He is encouraging whilst at the same time always reminding students about how to not make basic mistakes (which I do but I am getting there). Now I would like to learn Calculus and have been looking at various videos but this one is the best place to start. I have loved the Algebra course. Many thanks, John. Simple the best!
Right there with you. I’m 64 and did not do well in college calculus so I stopped at the AS degree. I’m retired now and have a full machine shop in my home shop. Trig is almost a necessary tool in the shop. Fortunately my buddy is a retired engineer with a BSME degree from U of F. He taught me more trig in one day than my professors in college and I’ve become very proficient with it. I welcome a math problem now! This video has motivated me to step it up and challenge my ridiculous fear of the Calculus. I’m sure it’s a combination of motivation on the student side and the ability of the teacher to communicate the concepts.
57 years old… I failed calculus in my freshman year of college. For nearly 40 years I’ve wondered about it but never learned it. Thanks for this amazing explanation!
SCHOOL SYSTEMS ARE 95 PERCENT SCUM, WHAT YOU LEARN FOR YEARS,YOU LEARN IN 5 -20 MINS ON TH-cam AND YOU WONDER WHAT WAS HAPPENING AND YOU REALISE SOMEONE WAS IN BUSINESS WITH YOUR TUITION FEE.
Which calculus you failed? There´s differential, Integral and Diff Equations as far as I know. Diff. Equations were extremely dificult for me and just mangaged to pass by memorizing everything, not by actually learning.
You have my total sympathy. And you know who are the major reason for you not understanding? Freaking mathematicians! masquerading as teachers! Especially calculus, which was mainly started by an engineer, Newton, and not some eggheaded number cruncher. I passed calculus in high school in England, but only worked out what it was really all about in second year fluid mechanics at university doing engineering. These people shouldn't be allowed anywhere near calculus - there are too many people like you (and to a lesser extent me) who weren't taught properly by mathematicians who haven't really got a clue how differentiation and integration are used in the real world!!
My breakthrough in calc came in my Physics class, when the instructor casually said, "Velocity is the first derivative with respect to time, acceleration is the second derivative of distance with respect to time, and the first derivative of velocity with respect to time." This brought it together for me, that, for example, if you integrate 60 miles/hour from 0 to 1 hour, you get 60 miles (to use the simplest example; one everyone is familiar with). I have pointed out that things like this are calculus that almost everyone does, and often frequently. It is a confidence builder for those who are starting with calculus.
Without reference to anything political, I found myself teaching a bit of discrete calculus when looking at daily COVID cases and deaths in the US and in Arizona where I live during the pandemic. The total cases (deaths) was the accumulation (integral) of daily cases, and the trend of daily cases, whether exponential, more linear, or flat, was the daily differences (derivative) of daily cases. In another case, I found myself arguing with a rocket scientist who continually made use of exponential curve fits to COVID data when the behavior of the curve near inflection points was not exponential at all. I found myself being somewhat mathematically insistent that the only curves which were their own integrals and derivatives are exponentials, which was a point lost on someone with more engineering-related training than that of pure mathematics. Except being in discrete time for the COVID cases as opposed to continuous time for Newron’s Laws of Motion wich you referred to, the same tools apply to both types of problems. And this makes sense when you think of position as accumulation (the integral) of velocity and acceleration the trend (or derivative) of velocity as well. Pandemics can have pedagogic advantages despite devastating real world effects. ;-)
One more step to the "adding area of rectangles" is needed to clarify what calculus does: As the number of rectangles increases, the size of the unmeasured space decreases. Calculus takes that to its final limit-: An infinite number of rectangles means the unmeasured space is infinitely small--i.e., the total space of the shape is accounted for.
At one point in my mechanical career, I had to implement a 4 to 20 milliamp transducer into a refrigeration unit that cooled wine for cold stabilization. This Refrigeration unit was a 300 hp Ammonia refrigeration unit. Basically, I had an electrically generated input (wine temp) and it had to go to a pneumatic output (ammonia temp). Well, to shorten the story, the 4 to 20 milliamp transducer was controlled via an "integral, derivative and proportional" (PID Loop controller) calculation that gave the program its parameters as to how to derive the steps needed for a smooth control of the output compared to the input temperature. I had a degree in electronics and had taken math from basic math to geometry to algebra I/II to trigonometry to my first semester of calculus. The mechanical engineer, who designed the system, was stumped as to how to control the wine to ammonia temps. The system "slammed" on and off. So, I told him I could do it with a 4 to 20 milliamp, PID loop controlled, transducer. Once, I dialed in the PID loop controller (4 to 20 milliamp transducer) parameters, the system worked like a charm. Math is how you figure out how to control a lot of things in life and I wish I had gone further than just one semester of calculus... I must edit. How did my electronic's training help??? I had to hook up a meter and found that the only way the PID loop controller would work is in the "reverse" setting, otherwise the system would slam on and off as if the controller was not in the system.
@@kc5402 How do you figure the PID loop controller's values? Proportional Integral Derivative (PID). You must use math to understand the input values that will give you the proper outputs... That is my point. If you have X (wine) temp. input and need a milliamp output to a pneumatic valve, you will have values that simple math can not solve, nor can you solve it with algebra. How do you "ramp" the steps in the program controller? It is not like carpentry where you have "rise over run" to get a stair built. How do you solve the needed amount of ammonia refrigerant to glycol coolant to wine temperature via a pneumatic valve controlling the glycol flow? That all relates to math and calculus is a valuable tool to understand complex mathematic problems. Hope that helps in my reasoning...
@@kc5402 Please look up what a Proportional Integral Derivative (PID) is and the Calculus it takes to get the answer. That should help. Without Calculus, you can not solve the PID loop controller's equation. Look it up. Hope that helps. Wiki has the equation written out for you to see the mathematical problem.
@@kc5402 BTW, here is the situation I was given. I have a 300 Hp ammonia compressor with a glycol filled heat exchanger. The glycol heat exchanger cools the wine that flows from a tank to a pump to another tank. Production says, "We need to bring the wine from 40 degrees down to 29 degrees for cold stabilization. We need the wine to flow at 300 gpm to meet production needs." We just got in this new ammonia system and its parameters were not set up. The compressor would "start and stop" if there was little or no load on it. So, there needed to be a way to get the wine temperature to 29 degrees without "slamming" the 300 hp screw compressor "on and off" which was the problem. The wine output had to control the pneumatic operated valve that controlled the flow of the glycol that cooled the wine. How do you do that? The glycol temperature was cooled by the ammonia refrigerant exchanger. That was my job, as a winery mechanic, to solve the refrigeration issue so that the wine production could meet schedule. The "3 fluids" had to be modulated so that the compressor would not "slam" on and off and the wine would not freeze nor would the glycol. Ramping the compressor up and down would not solve the issue that was tried by the Refrigeration engineer. His solution failed so they asked me (a mechanic with electronics and refrigeration background) to solve the problem. I put in a 4 to 20 milliamp, PID loop controller. And yes, I liked to work with my hands more than working behind a desk starring at paper or a computer screen ( in 1991)...
I got 85% in mathematics in post graduation. However I was Big Zero in getting why to use integration and calculas. Now I understood very well by simple example. Thanks a lot now I can teach my students very easily. Complicated subject is made very simple.
57 years ago first time for me in electronics school. I had to relearn it 11 years later when I became a controls engineer so as to use PID in writing software. Now at 75, here I am learning it again. I remember my first teacher was Mr. Rice and he had the same patient demeaner as you. Thanks.
I never get chance to learn calculus in high school, now finally I got an introduction lesson and completely understand it. Thank you for making the video and I have one less regret in my life!!👍👍👍🙏🙏🙏
I spent a semester doing calculus homework at work because our project was on hold. Have engineers around helped me ace it, however, not using it has made me forget most of it. This sounds SOOOO familiar! Thanks!
I am a software engineer now, and did Integration in College. But No one explained me Integration like this simple and fundamental way. Thank you very much.
I watched many TH-cam videos on this subject just to know what is calculus about, your video made it clear in a simple way. Thanks. Now I can buy a book to start learning calculus.
76 here...EE/MD. Lots of calculus & DE both as a student and tutor. Loved every bit of it and it changed our/my view of the world. As an example, it would be impossible to know things like E=MC^2, etc., etc., without calc/DE.
I wish you had been my calculus teacher. The first one was chinese, and taught it on trigonometry , which i had not had,, hopeless. I took the trig then back to calculus and got good grades but i so appreciate your simple explanation,, so very much better ,, and so much better to from a point of simple understanding and appreciation of it's value. I would have said i took it but did not really understand it nor its use, but this is so much better. I appreciate your attitude in helping to not make Math has to be terrible. Really good work.
It's been 37 years since I last had to solve an integral like in the thumbnail, but somehow the solution popped right into my head (14:31). I guess it stays with you for life. And for those who wonder "when would I ever have to do this kind of thing in real life," at least other than in academics and research, I can say in my engineering career you don't do it directly, but this helps you understand how numbers relate, and how physics and electronics relate to math. And that's how you come up with solutions to problems in electronics, mechanics and physics, and invent new things.
That was such a clear explanation, I am seventy one and haven't done calculus in decades. I wouldn't have remembered how to do that, but watching your video's that knowledge is slowly trickling back.
Thanks for taking the time to make this video, I'm 56 years old and am trying to re-learn math from foundation level up. I see the word calculus often and it still seems like something I could never understand. Perhaps I never will but at least your video gives me an insight and some hope that in the not too distant future I could begin studying calculus, as the simple concepts you outline are easy enough to understand.
I did differentiation , Integration a lot when I was 17-22 yrs age. scoring 9.4 in major tests back then. now I am 56 , forgotten 99% of it , but your ''infinite slices of thin rectangles'' opened it up wide again. would like to dwell into it again, it''s like a lost love. to me this lecture could have been packed in 3-4 min. max, but I understand , your audience is wide , just as wide as you opened it up for me here :)
You didn't explain how you dealt with x^2*d*x you merely left it up to the students imagination to figure out how you arrived at 3*3/3 - 2*3/3. This is the epitome of why so many students hate math.
I agree wholeheartedly, and then just ignoring the explanation of "dx" , ("which we don't have to talk about") seems to say it is not there for any particular reason, it needs as much explanation as the other parts.
He replaced the first x with the upper limit number 3 and he replaced the other x with the lower limit of 2. This gave him 3^3/3 - 2^3/3. Was that your question? There are a lot of moving parts to his explanation.
@@allanstewart5682I believe the dx is the delta for x, meaning the difference between the 3 and 2 limits. That’s why he subtracted two numbers. It’s really just an identifier and not really integral to the problem.
I was blessed to have an excellent calculus teacher, dragged me through all three semesters in college. Now of course 30+ years later I've forgotten 99.9% of it 🤣.
Great video 👍. It basically “basic” shows an example of a fundamental theorem. This vid is obviously aimed towards the end of high school to practically solve such problems. I saw a comment about still being confused. So, please contemplate trying to explain this theorem with the proof. It is a bit beyond high school, but, I think, still accessible to clarify the ‘add 1 to the exponent and divide by it.
i'm 36 and I graduated an Engineering course in college and I forgot how differential and integral calculus are. This is one great refresher for calculus! :D subscribed.
Thanks!. I took integral and differential calculus in college and am proud to say I got a C. My teacher gave me an option to bail out without an E, when I fell asleep in class to the lovely drone of his lecturing voice at the blackboard. But I got somehow a decent grade on the final exam. I hung in just because of my fascination :) I love the idea of it, and it inspired in me a powerful tool for creative logical investigation, e.g., where any topic can be considered by examining limits and boundaries. The bouncing back and forth from one extreme to another drives an exploration towards the truth.
You did a great job of explaining basic calculus. I’m 75 and was introduced to calculus at age 15. I wish you had been my maths teacher all those years ago, because I really understood for the first time! 🇬🇧
You really should explain the "dx" symbol. I was an outstanding math student in high school. Then I got to college and had to take calculus, and failed miserably. I tracked it back to this symbol. My lack of understanding of this symbol interfered with my abili8ty to do calculus. And I've never found a satisfactory explanation for it since.
Here's why it didn't make sense to you (or anyone else with a brain)... Calculus Foundations: Contradictory: Newtonian Fluxional Calculus dx/dt = lim(Δx/Δt) as Δt->0 This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale. Non-Contradictory: Leibnizian Infinitesimal Calculus dx = ɛ, where ɛ is an infinitesimal dx/dt = ɛ/dt Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.
Well I guess the closest thing I can describe it to is like (1/(infinity))*x. It is an infinitely small of part of x, there are infinite of these parts. When you integrate you add all these infinite parts to get x. For anything higher than integration of dx I don’t know. I just solve without thinking about it since it isn’t in my syllabus. The best thing I can think of is when you integrate f(x)dx, you take a tiny part of f(x) and add it another infinity of itself(ignore this I don't really know).
I’m the same. I never found someone who could explain it to me. I suspected my maths teacher didn’t understand it either. It’s like floating rate notation. People use it and just accept they don’t understand it.
I am an artist. This post looks to be a good one for a non-math person. I am getting a headache right now, but I will revisit your post later when I have the time to focus my brain. (I was a senior in high school in the junior algebra class.) I am good with geometry, but, this a level above. Thanks.
I like the way you prompt your discussion. It's so cozy and not intimidating at all. Like you are coaxing us all to sleep soundly with dreams full of numbers. And after I wake, everything in reality is great with numbers
Thanks, this lesson was just about the level where I dropped out of my communications electronics trade back in the 70's. Thing was some teachers/lecturers can teach and some simply can't. In my case it was simply me ! I think if I was just a little less burdened with life I could take up an interest, not sure how far I could go but maths has ALWAYS fascinated me. It's predicted so much, so many discoveries because maths showed the way. I enjoyed your lesson and at least now I will remember what that symbol is..integration, I hope.
I did learn in my pre degree ( 1978) ..felf more comfortable in solving differential calculus problems.. Thanks for your simple practical application of integral cals..presently working in an Engineering & Cont firm as Finance Manager at Muscat...
Great illustration of why so many struggle with math. Hey let's pull the number one out a hat and add it to the exponent with no accompanying explanation....etc.
Man thank you. Maths is like a language, you either speak it or do sign language. Those who can speak maths remain the best in maths. And you sir are one the best teachers.
One think I thought was always lacking in my math education was that teachers would never explain what the application is for what was being taught...for many people, that would be very helpful in learning how the material could be applied. This video is a great example of this. There is zero mention of what this concept can be used for. If teachers did more of this, it could inspire many more people to be interested in math I think. what would be examples of x and y and how does that area under the curve telll us something.
The way I explain what integration is and its application is simply as accumulation. For example, if I accelerate at a certain rate, what speed will I be doing after 10 seconds? Accumulating acceleration gives you speed - the integral of acceleration is speed. What happens if I then accumulate the speed I got in the previous step, how far have I gone? Distance is the integral of speed. (And for derivatives, going the other way of course) Other applications, e.g. geometric, literally a use of measuring the area under a curve - imagine I'm designing a fuel tank that fits in an aircraft wing and has a constant cross section, which is curved. If I know what this curve is, I can turn it into a function, find the integral (the area under this curve), and work out the cross section and then the volume by multiplying that by how long the fuel tank is. I can easily use a little bit of algebra so I can have a specification eg. the customer wants a 100 litre tank, and its cross section will be this, how long will this tank need to be? Then there's things like PID controllers, where I stands for Integral and D stands for derivative, both from calculus - it mixes proportional, integral and derivative values to produce a control signal, e.g. to keep an oven at a constant temperature. The proportional is how far off are we right now, the integral is how much error we've accumulated, and the differential is the rate of change. These are some simple real world examples that should really be introduced because as you say, in a vacuum, young students especially will wonder what it's for!
@mikeearls126, as a senior in high school in 1969, I thought the same thing. My councilor advised me to take Physics along with Trig/Advanced Math(Calculus was contained in this course). Everything @74HC138 mentions above along with light, particles, waves, electricity and many other things were taught in the Physics course. All the math needed for all of this was taught in both classes. Sometimes we learned what we needed in the math class but other times, we didn't get that far yet so the Physics professor taught us what we needed. To me, the combination kept me more interested than I may have been.
Exactly. We were taught it from a strictly math viewpoint, with no idea of what we were actually doing. IN the exams we crashed with integration or differential calculations. We didn't understand the practical application, so it was just mumbo jumbo to us. If the questions were framed about how much money etc etc, or how many apples you could steal from a 8 foot tall tree etc etc given what ever the circumstances were, we were all experts on that sort of calculation!!! For some unknown reason, we all understood money and stealing apples from neighbouring orchards. And we looked like angels... butter wouldn't melt in our mouths.....
I agree. If I had been explained how the maths was used practically it would have been more interesting. I did a plumbing course a few years ago and suddenly the maths I had learnt at school became more interesting and I also began to understand why some formulas were as there were.
Thanks a lot for this amazing explanation. I did calculus I to III in campus 10 years ago but I survived by memorizing things instead of really understanding the concepts that you clearly explained here.
4:00 You have to use the integration formula to solve for area problems with irregular shapes because integrals provide the best estimation for the area rather than just guessing and checking. So, rather than using a million little rectangles, the integral formula works to get the perfect area
BEING A MECHANICAL DESIGN ENGINEER, I STUDIED CALCULAS HOWEVER I NEVER USED IT & EVEN FORGOT ALMOST ALL OF IT, EVEN THOUGH I RELISHED IN ALGEBRA, TRIGONOMETRY & GEOMETRY WITH HIGH MARKS DURING THE 1960'S. BUT SEEING THE BASIC CONCEPTS ALL THESE YEARS LATER REFRESHED MY MEMORY. SO THANK YOU FOR YOUR VIDEO.
I loved calc in high school. Flow rates of change, position/velocity/acceleration, area/volume. Just working through orders of dimension was fascinating to me. Then I went to college, had a few lousy professors, and became a business major where division is considered complicated.
I'm 67, recently retired, and I failed calculus 3 seperate times after High School over about 15 years. The Calculus Book for Dummies was my fourth attempt around 2010. I did understand with following the book. But hand me a calculus problem and I'd never get it. Learning a few basic rules you covered here gave me a better understanding. And I have basic geometry like in your example to approximate the area under a curve. I understand fractions very well. In fact my high school chemistry teacher taught us 2 ways to do Stochiometry (balancing chemical equations) one with fractions, and the long way. Guess which one I went with.
The old saying goes " use it or lose it." Something complex like the language of math is if your not actively using it every day it slips back into the abyss !
It’s like Algebra and Logarithms, it’s not relevant to my day to day life, but in certain professions like computer science and engineering, it will be. The trouble is that we weren’t taught how mathematics are used in the real world away from the classroom.
It’s up to us to use these tools in everyday life. That is the value of education. It teaches you to think. Application to the real world was never going to be easy.
My biggest gripe is at the very beginning of the class No one explained why it was important to know how to do it that would have interested me more and I would have paid closer attention. I have taught various classes once I was older and I always explain in detail what's the whole point of the class in the beginning. It sets the right tone and I can refer back to what I told him the very first day as we go along so that by the end of the class they have the best chance of understanding the material and the reason to invest so much time and energy to understand what it means.
62 here . Back in university, I got straight A's in the complete course in single-variable calculus, but that's as far as I went, and because I never had to use it in real life (despite a 40-year career as a software engineer), the knowledge atrophied to where I couldn't do it to save my life. Thanks to channels like this, It's all coming back to me ... but I still haven't yet needed it in my daily life. 😕 Maybe I'll pick up a retirement hobby that requires it ... Any suggestions?
the dx is simpy abbreviation from differential-x and yes the integration is 1-power up, and differentation is 1-power down. differential is slope or m; example: dif. x² = 2x (the number 2 become multiplier/moved to the front, the power 2 reduced by 1), if reversed, int. 2x = x², or int x = x²/2 (the power up +1, the number of power become divider) hence int x² = 1/3 x³, int x³ = 1/4 x⁴ ... and so on... that's what i can recall from my highschool in 80s 😂
I was an AMIE student in the early eighties preparing for Part I since I was a Diploma holder at that time.I vigourously prepared for Maths took tuition from one professor of Maths.Even though I could not succeed in my two attempts I still admire both Differential and Integral Calculus.I still have one book authored by Professor Hardy.
This was awesome. One question. How did you know to add 1 to the exponent? Is it because the difference between 2 and 3 is one? In other words, if we were going from 2 to 6, would you add 4?
I will not bother proving why it is so but it is always "+ 1" (for functions similar to the one used in the example which f(x) = x^2). Going from 2 to 6 would not change that. The final answer would change though because it would be equal to this: 6^3/3 - 2^3/3 which is equal to 6x6x6/3 - 2x2x2/3 = 216/3 - 8/3 = 208/3 = 69 1/3. (The reason for the "+1" would be too long to explain... sorry!) By the way if the curve's formula is f(x) = x^5 instead of the example shown, the integral would again have a "+ 1". It would be x^6/6... being between 2 and 3 or whatever two values on the x-axis doesn't play any role in this. Hope this helps a little bit!
I think this answer could use Espo to bang the puck back to Henderson. The “+1” is part of what’s called “the Power Rule”, one of the many “tricks” (shortcuts) we use in calculus. @TeamCanada72 is correct, the proof is long and abstruse, but there are numerous explainer videos available on TH-cam and elsewhere
Thank you for An excellent presentation. I am a retired engineering doctorate. This reminds me of my college times where we had to find the distribution of charge in a plane with a singularity represented by a pin hole knocked into the plane. I enjoyed returning back to basics. Hope the young generations have teachers that explain maths in such a simple way as you did.
Recall the original problem integrate X²dx with respect to x with limits set at 3 and 2. After integration, it will be x³/3 + C (constant). Substitute the limits 3³/3 + 3 - (2³/3 + 2) and so on. Loved this math and differential equations. It has been around 45 years since college.
It’s even using some algebra principals on how do we deal with fraction. It’s using every other math that we learned throughout the years and putting it all together.
Back in my University days (1969 - 1973), there was no degree in Computer Science or even a department focused on computers - but wanting to get into compute programming, I could only go with a BS Math and take all the computer courses offered by the Math Dept. I had to suffer through 16 semester hours of Calculus. I survived it, barely. Got my degree, and then built a successful career as a Software Engineer, never having to do an integral or derivative, ever again. Thank God! I have to add that I never had an instructor explain the very basic concepts of Calc like you have just done.
I wish I could say the same. I started my own Electronics engineering company totally weak on calculus and short-cutted and estimated my way though. I knew what the results should be so touched on it enough to get by but I wish I'd have seen this, it would have made my life a lot easier!
Archimedes already developed infinite summation (integration) but unfortunately was killed by Roman rabble. It was rediscovered by Isaac Newton, 2000 years later.
Invented by two independent men who didn't know each other : Gottfried Leibniz and Isaac Newton. Today we use the Leibniz notation because it is more elegant.
@@paulanizan6159 interesting you should mention that long unnecessary gap in our intellectual history. For my history senior thesis in spring 1976 I wrote essay stating that there was 1800 years between Archimedes and Galileo in terms of the development of math. I blamed the Church mostly but my history prof taught medieval euro history and did not like my assertion. I stick by it till this day. The Church and Roman Empire squashed the logical progression of math for 1800 years. And I asked at end of essay "Where would we be today in progress of physics had a Galileo appeared in 100 BC?"
My experience with calculus is that most of its concept are quite simple, but quickly result in very complex algebra. There is also the logic of how to apply the rules you know to the problems at hand.
that big "S" means sum up... from 2 to 3 integrate means to add a power.. i.e. x^2 becomes x^3 here... (1/3)x^3 + C at x=3 (1/3)(3^3) + C =9 + C at x = 2 (1/3)(2^3) + C = 8/3 + C (9+C) - ((8/3)+C) =9 - 8/3 =(27-8)/3 =19/3 = 6&(1/3)
That's right he forgot the constant entirely - tho those cancelled since it was a definite integral, still it should have been included as a general case
Your explanation and introduction into calculus is brilliant!! If I had a math instructor like you back in high school and freshman college, I might have had an increased interest and done better. Of course, paying attention and asking for help could have worked too….😊. Just want to say thanks. Love TH-cam. Damn, I’m fifty years too late.
TabletClass Math, could you please explain how 'x squared dx' becomes 'x cubed divided by 3'? Am I corrected that 'x squared dx' is the first derivative of 'x cubed divided by 3' ? Thanks A thought, does x**3 dx become x**4/4?
Just a story of my sophomore year at Aalto university (Espoo, Finland). It was a year of 1983 and I was doing serious math. One home assignment was to calculate of an area of 3-d object. It was defined with minimal info. It took me some 3 hours to make a mental picture what was the shape of the object. It was late Friday evening and my room mate was preparing to go out for fun. My fun was math. When I figured out that it was a banana defined with math formulas. Then it was easy to calculate the defined surface area. I was happy.
Oh dear! You skipped the most important point to understand.Yes, we are adding up the areas of the whole series of rectangular shapes. But each of the tiny rectangles has a height of x^2 and a tiny width of delta x, as delta x approaches zero. There is no understanding of this problem or any other without understanding dx represents the width of each rectangle and the concept of the limit as delta x approaches 0. Without this understanding it just becomes memorizing another formula.
It doesn't matter... as someone who really, really struggled with math in high school (and continues to do so), understanding this watered-down concept was a revelation ... I actually understood what what going on and my number-phobic brain stepped up a gear.
When I was in school, none of this made sense for me and I was hesitant to ask questions. I am 43 now and trying to understand it. Thank you for the great explanation 👍🏻
You're smart and you're way ahead of my generation I went to high school in the 1970s algebra itself wasn't even a mandatory class. You didn't have to take it😊
I cannot say this Ralph. I heard that university students can not even write cursive or do some simple math in their heads but need a calculator for the simplest calculation which I do in my head in a split second.
@@paulanizan6159I know somebody who has a degree in sociology, he needs a calculator just to do basic math he's terrible in math and he'd be the first one to admit it
When I was first learning calculus almost 50 years ago, my big stumbling block was trying to figure out why I should care about the area. I was more interested in the function(s) that bounded the area. Honestly, reckoning speed and acceleration was much clearer since I had real-world examples that could be applied much more easily than area. Either way, certainly at the beginning, I just applied the laws for integrals and derivates to get an answer without actually understanding the applicability of what I was doing. It was just math problem -> math answer. At some point it was more of an "a-ha!" moment than my teachers or professors explaining it so I really understood it. (I liked Jaime Escalante's, "I don't have to make calculus easy because it already is," from Stand and Deliver. And it kind of is...once you "get it".)
Calculus Foundations: Contradictory: Newtonian Fluxional Calculus dx/dt = lim(Δx/Δt) as Δt->0 This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale. Non-Contradictory: Leibnizian Infinitesimal Calculus dx = ɛ, where ɛ is an infinitesimal dx/dt = ɛ/dt Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.
Thank you for giving me some start of an understanding of how it works. I missed a few days at secondary school and never caught up, but have always meant to look at it again. I had forgotten even that y=x squared even. It’s almost like magic and I’m left wondering how on earth anyone could have come up with all this in the first place!
What a fantastic little film. I 'learned' calculus at 40 years ago in a very dry old fashioned way. So I could do the maths 'algorithmically' but never 'got it'. This was fantastic. A description of 'why' the rules are what the are would be more challenging to explain but would be appreciated.
I have always wanted to conquer my fear of math and learn more math than I did, including understanding the concepts behind the equations and formulas. Maybe this is beyond the basics, but I was disappointed when John didn’t explain why 1 was added to X squared. Without any explanation for why this was done, I felt left in the dark.
I've been the university's top dog in calculus - differential and integral, it helped me understand where the formulas came from. And now after 20 years, seeing this video, I've completely forgotten about it, maybe because after college, I could not apply calculus in every day life.
You made the explanation so easy, that each and everyone on the face of the earth can understand it! Your explanation is the real life example of George Polya's problem solving process should look like plus pedagogical talant! Congrats!
I took 3 semesters of calculus in college, but that was 40 years ago. I also began to see it as a language to describe classical physics and engineering. Fun to review. Thanks. However I never used in my various careers.
That's exactly what frustrates me. What's the reason behind that formula? Yes, it's easy to memorize this formula but curious minds must know how did the mathematicians conclude this to be the exact answer. I'll search a bit on that. Will get back to you if and once I find something of value.
Refer labanitz theorem from which it’s derived . Very basic is sequence and series, continuity of function and limits how can he explain everything . He made the videos for how irregular areas are calculated . In a rough way
Thank you for opening up a world to me today. Despite a zillion ads that played throughout the video, I really feel like I learned something very important today. Thank you.
An explanation that leaves too much unexplained. “Here we add 1.” Uh (raises hand) why do we add 1? Why do we add anything at all? Why not 5? Is 1 a personal friend of the instructor? Did 1 pay an endorsement fee? What’s wrong with 5? Has 5 been blacklisted? Is there some unseen prejudice at work? Is there no justice? I don’t think you should call it a “basic” lesson or introduction if you are going to gloss over a fundamental and integral part (ha, see what I did there?) or assume some specific insider knowledge on the part of the audience. IMHO
The title only says “basic” calculus. Which it is a basic calculus operation for teaching purposes. He also mentions he has more content available if you want to go deeper into any lesson. It’s good that you question, but even better is to be proactive on your pursuit of knowledge. Keep learning 🤙
I did the chatgpt thing: “Adding 1 When Integrating: When you integrate (find the antiderivative), you add 1 to the exponent and then divide by the new exponent.”
I did Calculus at University for a while but never completely understood it. I think this has helped to explain something I probably didn't know or forgot. I would be interested to know why the integral works and relates to the rectangles.
For you guys who don't understand the very first question of "Why would I ever need to know the area of under that curve?" Many fields using these cartesian quadrant. From engineering, economics, coding, marketing, statistics, music, or even today's influencers. All of them need a tool to analyze stuff accurately. Calculus simply a way to operate one of the tools. Like, for example the Demand Supply curves in economics often need to find area under those. Which involving prices of products, or profits of that products, or decisions regarding that products, etc. Which ultimately affecting us all in the end of those. Any Cartesian Quadrant will involve curves, then curves will involve calculus. Basically, everything involve calculus if you delve deep. Scribble a random line in the quadrant and there will be a "name" for that scribbles (in this video, the curve's name is X^2). There is a way to find your random scribbles' "name", there is a way to find the area under that scribbles, and then there is a way to find *VOLUMES* when you start involving the next axis of the "Z" axis (add another elongated "S"). So basically, a line into 2D area, then area into 3D volume. Then those academics that using triple fold of those elongated "S" of integrals already talking in the next dimensions of axises (whats the plural of axis?). We cannot see it, but we still can calculate it. Its like a blind cant see the sun, but can measure its heat. All of that for a simple rule of "Just add the power by one and divide with that." School back then were simply trying to give us the manuals to use this tools, in hope that we can use it too. Just like my daughter's play group trying to teach my daughter how to use toll called pencils, my uni tried to teach me how to use tool called calculus.
90 years old.There was a 70 years ago I could have solved that problem but I don't think I really understood it. Your method seems to be better than whatever it was when I took calculus.
I'm 71 and did a lot of calculus while studying for my master in engineering. Well, now, 50 years later, I don't remember anything about it. Thank you for this little insight. I enjoyed it.
Lot of friends I know just passed the math classes and went to work in a shop.🤓
I'm 72. Welcome to my world. I have two entering degrees, one in electronics and the other in electrical engineering. I was once buff, pretty (this is a joke; I'm a male, lol), and bright at least once. I have since forgotten most of this stuff. Computers came to the rescue in the 90s. High-end calculators, like those on smartphones, took care of this for me. I have a learning disability-bad memory. I'm not able to remember formulas. But nothing I've built or repaired blew up.
I took the 3 required semesters of Calculus in College for my Comp Sci degree and never used it again(tiny bit in my MBA coursework). Just here for a refresh 40 years later. I just remember it being rigorous with 2-3 hours of homework each time we met. Now that I'm retired, I get to learn just for the fun it. Onward into Python now.
I’m also 71 with a civil engineering degree and I can’t recall any of my calculus and differential equation classes but enjoyed this video.
I'm 76 with a Bachelors degree in mechanical engineering technology. This brings back a lot of memories of my technical calculus classes. I still go over my calculus papers and books.
I'm 69 1/2 years old and hated mathmatics since childhood. I struggled through high school and college with the bare minimum courses in algebra and geometry just to get through school. Thank-you for showing me in your kind calm way how this works. I won't become an Einstein, but at least I can go to my grave saying that I actually understood a basic look at calculus and understand why it is important to our civilization. Thank-you for that, sir!
I'm 67, and have a degree in Aeronautical Engineering. I struggled thru calculus in college back in 1975-76 and after that year, I never touched Calc again until a few minutes ago. I think I agree 100%$ with you!
@@QuantumRift I'm an idiot, but having calculus described as finding the area of weird shapes, learning that you don't use it in aerospace engineering is mind boggling to me
I studied Calculus in college, then on Engineering..scored distinction. Then worked...worked ...then worked in Engineering field.. Retired now. Still waiting for the day to use Calculus knowledge.
well I don't know. I can follow it but there's still a lot that is all Greek to me. I guess like he says if you know the rules you just apply them and you got the answer-but that's where I'm lost. I don't understand why and where you get the stuff following the integral sign. A bit of it maybe like the 2 squared and the 3 squared but that's it.
I read this and had to recheck your name, thinking, 'did I post this sometime in the past and forgot?'
65 1/2 here and exactly the same (including two hot summer remedial sessions in math). but after my recovery from high school I regained an interest and went back to night school for another go, at pre-calc and calc.
Failed but completed and it rekindled something teachers had nearly killed off years ago.
A good presentation. I could've used some of the gaps filled in ("so, that works for THAT shape, but what about ....") but I get who the target audience is and I'm still in that group. ; )
I'm a short distance to my 79th. but still following maths in U-tube. I have already learned matrix and now I'm glad to start Calculus with you. Never did in my O-level back in the late 60s. Thanks from Zanzibar
Good for you!!!! I'm ten years behind you and I cannot stay away from this !!!
It's amazing how the things that sucked balls for us at high school interests us as we get older. Maths and history for me.
Maybe we're aiming this whole education thing at the wrong age demographic! 😂
Good man, excellent work.
You are admirable
@@theshadypilot Crazy huh? 🤣🤣🤣
You explained the material so well and in an easy way to understand! A big thank you! 😊
66. I taught calculus and other math for 35 years. This is well done, I've done this material dozens, maybe over a hundred times.
He lost me at 2+1 and some calculus rule and I took Calculus in ~1987
@@GTKJNow yeah, at that stage he just showed a rule without any explanation behind that rule - so at the end of the day, it didn't really shed that much light on calculus other than the titular concept of summing up the areas of thin rectangular strips beneath a segment on a curve
How many 50+ people are watching this video?😊
I lived half of my life as a soldier and the remaining in medical field, just curious to know if there are other people like me who missed math. 🙂🙏
Me, and I have an Academic Minor in Mathematics.
Me too
Me too
I’m only 49, took calc classes back in the day and forgot it all. Here for a refresher.
I am 60. This video is a reminder how incompetent my teachers were when I grew up .
I just wanted to say a Big Thank You to John, so I hope he reads this.
I am just finishing his Algebra 1 course and completed his foundation course before that. At the age of 62 I decided to confront my fear of maths, it was some unfinished business since I realised why I had developed a belief that I couldn’t do it at high school and that I should be able to overcome that. For the past year I have worked hard and have really felt that John was at my side and on my side - even though he does not know me. I have understood everything and to my surprise and delighted I have found that actually, yes, I can do math and moreover I love it!
I love the way he makes such an 13:37 effort to explain things clearly and methodically. He is encouraging whilst at the same time always reminding students about how to not make basic mistakes (which I do but I am getting there).
Now I would like to learn Calculus and have been looking at various videos but this one is the best place to start.
I have loved the Algebra course.
Many thanks, John. Simple the best!
The owner of this channel has math courses he teaches?
Right there with you. I’m 64 and did not do well in college calculus so I stopped at the AS degree. I’m retired now and have a full machine shop in my home shop. Trig is almost a necessary tool in the shop. Fortunately my buddy is a retired engineer with a BSME degree from U of F. He taught me more trig in one day than my professors in college and I’ve become very proficient with it. I welcome a math problem now! This video has motivated me to step it up and challenge my ridiculous fear of the Calculus. I’m sure it’s a combination of motivation on the student side and the ability of the teacher to communicate the concepts.
@@rubencollazo8857 yes, TCMathAcademy online
Love it. So inspiring!
@@charlieromeo7663 bless you. Keep inspiring.
57 years old… I failed calculus in my freshman year of college. For nearly 40 years I’ve wondered about it but never learned it. Thanks for this amazing explanation!
SCHOOL SYSTEMS ARE 95 PERCENT SCUM, WHAT YOU LEARN FOR YEARS,YOU LEARN IN 5 -20 MINS ON TH-cam AND YOU WONDER WHAT WAS HAPPENING AND YOU REALISE SOMEONE WAS IN BUSINESS WITH YOUR TUITION FEE.
Which calculus you failed? There´s differential, Integral and Diff Equations as far as I know. Diff. Equations were extremely dificult for me and just mangaged to pass by memorizing everything, not by actually learning.
You have my total sympathy. And you know who are the major reason for you not understanding? Freaking mathematicians! masquerading as teachers! Especially calculus, which was mainly started by an engineer, Newton, and not some eggheaded number cruncher. I passed calculus in high school in England, but only worked out what it was really all about in second year fluid mechanics at university doing engineering. These people shouldn't be allowed anywhere near calculus - there are too many people like you (and to a lesser extent me) who weren't taught properly by mathematicians who haven't really got a clue how differentiation and integration are used in the real world!!
@@stevedavidson666 you don't learn from teachers. just buy one or two good books.
Mee too
My breakthrough in calc came in my Physics class, when the instructor casually said, "Velocity is the first derivative with respect to time, acceleration is the second derivative of distance with respect to time, and the first derivative of velocity with respect to time." This brought it together for me, that, for example, if you integrate 60 miles/hour from 0 to 1 hour, you get 60 miles (to use the simplest example; one everyone is familiar with).
I have pointed out that things like this are calculus that almost everyone does, and often frequently. It is a confidence builder for those who are starting with calculus.
And set the derivative to zero, and you find the peak (or minimum), often useful.
Without reference to anything political, I found myself teaching a bit of discrete calculus when looking at daily COVID cases and deaths in the US and in Arizona where I live during the pandemic. The total cases (deaths) was the accumulation (integral) of daily cases, and the trend of daily cases, whether exponential, more linear, or flat, was the daily differences (derivative) of daily cases.
In another case, I found myself arguing with a rocket scientist who continually made use of exponential curve fits to COVID data when the behavior of the curve near inflection points was not exponential at all. I found myself being somewhat mathematically insistent that the only curves which were their own integrals and derivatives are exponentials, which was a point lost on someone with more engineering-related training than that of pure mathematics.
Except being in discrete time for the COVID cases as opposed to continuous time for Newron’s Laws of Motion wich you referred to, the same tools apply to both types of problems. And this makes sense when you think of position as accumulation (the integral) of velocity and acceleration the trend (or derivative) of velocity as well.
Pandemics can have pedagogic advantages despite devastating real world effects. ;-)
Thanks!
One more step to the "adding area of rectangles" is needed to clarify what calculus does: As the number of rectangles increases, the size of the unmeasured space decreases. Calculus takes that to its final limit-: An infinite number of rectangles means the unmeasured space is infinitely small--i.e., the total space of the shape is accounted for.
At one point in my mechanical career, I had to implement a 4 to 20 milliamp transducer into a refrigeration unit that cooled wine for cold stabilization. This Refrigeration unit was a 300 hp Ammonia refrigeration unit. Basically, I had an electrically generated input (wine temp) and it had to go to a pneumatic output (ammonia temp). Well, to shorten the story, the 4 to 20 milliamp transducer was controlled via an "integral, derivative and proportional" (PID Loop controller) calculation that gave the program its parameters as to how to derive the steps needed for a smooth control of the output compared to the input temperature. I had a degree in electronics and had taken math from basic math to geometry to algebra I/II to trigonometry to my first semester of calculus. The mechanical engineer, who designed the system, was stumped as to how to control the wine to ammonia temps. The system "slammed" on and off. So, I told him I could do it with a 4 to 20 milliamp, PID loop controlled, transducer. Once, I dialed in the PID loop controller (4 to 20 milliamp transducer) parameters, the system worked like a charm. Math is how you figure out how to control a lot of things in life and I wish I had gone further than just one semester of calculus... I must edit. How did my electronic's training help??? I had to hook up a meter and found that the only way the PID loop controller would work is in the "reverse" setting, otherwise the system would slam on and off as if the controller was not in the system.
And what does that have to do with calculus? Your long story doesn't seem to have any connection with the video!
@@kc5402 How do you figure the PID loop controller's values? Proportional Integral Derivative (PID). You must use math to understand the input values that will give you the proper outputs... That is my point. If you have X (wine) temp. input and need a milliamp output to a pneumatic valve, you will have values that simple math can not solve, nor can you solve it with algebra. How do you "ramp" the steps in the program controller? It is not like carpentry where you have "rise over run" to get a stair built. How do you solve the needed amount of ammonia refrigerant to glycol coolant to wine temperature via a pneumatic valve controlling the glycol flow? That all relates to math and calculus is a valuable tool to understand complex mathematic problems. Hope that helps in my reasoning...
@@kc5402 Please look up what a Proportional Integral Derivative (PID) is and the Calculus it takes to get the answer. That should help. Without Calculus, you can not solve the PID loop controller's equation. Look it up. Hope that helps. Wiki has the equation written out for you to see the mathematical problem.
@@kc5402 BTW, here is the situation I was given. I have a 300 Hp ammonia compressor with a glycol filled heat exchanger. The glycol heat exchanger cools the wine that flows from a tank to a pump to another tank. Production says, "We need to bring the wine from 40 degrees down to 29 degrees for cold stabilization. We need the wine to flow at 300 gpm to meet production needs." We just got in this new ammonia system and its parameters were not set up. The compressor would "start and stop" if there was little or no load on it. So, there needed to be a way to get the wine temperature to 29 degrees without "slamming" the 300 hp screw compressor "on and off" which was the problem. The wine output had to control the pneumatic operated valve that controlled the flow of the glycol that cooled the wine. How do you do that? The glycol temperature was cooled by the ammonia refrigerant exchanger. That was my job, as a winery mechanic, to solve the refrigeration issue so that the wine production could meet schedule. The "3 fluids" had to be modulated so that the compressor would not "slam" on and off and the wine would not freeze nor would the glycol. Ramping the compressor up and down would not solve the issue that was tried by the Refrigeration engineer. His solution failed so they asked me (a mechanic with electronics and refrigeration background) to solve the problem. I put in a 4 to 20 milliamp, PID loop controller. And yes, I liked to work with my hands more than working behind a desk starring at paper or a computer screen ( in 1991)...
@ababbit7461 why didn't you just ask chatGPT?
I got 85% in mathematics in post graduation. However I was Big Zero in getting why to use integration and calculas. Now I understood very well by simple example. Thanks a lot now I can teach my students very easily. Complicated subject is made very simple.
Very good teacher. We need more teachers like you in schools. Keep up the great job.
Excellent re-introduction of integral calculus after 60 years. Thank you.
.....wow, just did the same thing. only been 44 for me however!
And for me at age 76
@@rezamohamadakhavan_abdolla8627 🤣🤣🤣. Is it only over 60 year old can join this class? Old buddy classroom.😂😂😂
Half way through I said " why bother"? 😂
57 years ago first time for me in electronics school. I had to relearn it 11 years later when I became a controls engineer so as to use PID in writing software. Now at 75, here I am learning it again. I remember my first teacher was Mr. Rice and he had the same patient demeaner as you. Thanks.
I never get chance to learn calculus in high school, now finally I got an introduction lesson and completely understand it. Thank you for making the video and I have one less regret in my life!!👍👍👍🙏🙏🙏
Enormously helpful. I struggled with Integrals in college and this fixed something in 10 min what I couldn’t grasp in 2 semesters. Thank you!
You had inept teachers.
The biggest problem with most math is the poor teaching quality, not the students' intelligence.
No one has ever explained WHY AND WHAT we were doing like you are explaining it now. This makes so much sense. Thank you.
I spent a semester doing calculus homework at work because our project was on hold. Have engineers around helped me ace it, however, not using it has made me forget most of it. This sounds SOOOO familiar! Thanks!
I am a software engineer now, and did Integration in College. But No one explained me Integration like this simple and fundamental way. Thank you very much.
I watched many TH-cam videos on this subject just to know what is calculus about, your video made it clear in a simple way. Thanks. Now I can buy a book to start learning calculus.
76 here...EE/MD. Lots of calculus & DE both as a student and tutor. Loved every bit of it and it changed our/my view of the world. As an example, it would be impossible to know things like E=MC^2, etc., etc., without calc/DE.
I wish you had been my calculus teacher. The first one was chinese, and taught it on trigonometry , which i had not had,, hopeless. I took the trig then back to calculus and got good grades but i so appreciate your simple explanation,, so very much better ,, and so much better to from a point of simple understanding and appreciation of it's value. I would have said i took it but did not really understand it nor its use, but this is so much better. I appreciate your attitude in helping to not make Math has to be terrible. Really good work.
It's been 37 years since I last had to solve an integral like in the thumbnail, but somehow the solution popped right into my head (14:31). I guess it stays with you for life. And for those who wonder "when would I ever have to do this kind of thing in real life," at least other than in academics and research, I can say in my engineering career you don't do it directly, but this helps you understand how numbers relate, and how physics and electronics relate to math. And that's how you come up with solutions to problems in electronics, mechanics and physics, and invent new things.
That was such a clear explanation, I am seventy one and haven't done calculus in decades. I wouldn't have remembered how to do that, but watching your video's that knowledge is slowly trickling back.
Thank you for your clear and simple way of teaching a concept, something at which the majority of teachers, at any school level, fail!
Thanks for taking the time to make this video, I'm 56 years old and am trying to re-learn math from foundation level up. I see the word calculus often and it still seems like something I could never understand. Perhaps I never will but at least your video gives me an insight and some hope that in the not too distant future I could begin studying calculus, as the simple concepts you outline are easy enough to understand.
I'm 79. Getting encouraged by maths lovers like you. I'm equally still learning the subject. Greetings from Zanzibar
I did differentiation , Integration a lot when I was 17-22 yrs age. scoring 9.4 in major tests back then. now I am 56 , forgotten 99% of it , but your ''infinite slices of thin rectangles'' opened it up wide again. would like to dwell into it again, it''s like a lost love. to me this lecture could have been packed in 3-4 min. max, but I understand , your audience is wide , just as wide as you opened it up for me here :)
You didn't explain how you dealt with x^2*d*x you merely left it up to the students imagination to figure out how you arrived at 3*3/3 - 2*3/3. This is the epitome of why so many students hate math.
I agree. I hated math for this exact reason. I wished he had explained that. I was following him up to that point.
I agree wholeheartedly, and then just ignoring the explanation of "dx" , ("which we don't have to talk about") seems to say it is not there for any particular reason, it needs as much explanation as the other parts.
I also agree! He skipped a big step here. Poor explanation for sure
He replaced the first x with the upper limit number 3 and he replaced the other x with the lower limit of 2. This gave him 3^3/3 - 2^3/3. Was that your question? There are a lot of moving parts to his explanation.
@@allanstewart5682I believe the dx is the delta for x, meaning the difference between the 3 and 2 limits. That’s why he subtracted two numbers. It’s really just an identifier and not really integral to the problem.
I was blessed to have an excellent calculus teacher, dragged me through all three semesters in college. Now of course 30+ years later I've forgotten 99.9% of it 🤣.
Great video 👍. It basically “basic” shows an example of a fundamental theorem. This vid is obviously aimed towards the end of high school to practically solve such problems. I saw a comment about still being confused. So, please contemplate trying to explain this theorem with the proof. It is a bit beyond high school, but, I think, still accessible to clarify the ‘add 1 to the exponent and divide by it.
i'm 36 and I graduated an Engineering course in college and I forgot how differential and integral calculus are. This is one great refresher for calculus! :D subscribed.
Thanks!. I took integral and differential calculus in college and am proud to say I got a C. My teacher gave me an option to bail out without an E, when I fell asleep in class to the lovely drone of his lecturing voice at the blackboard. But I got somehow a decent grade on the final exam. I hung in just because of my fascination :) I love the idea of it, and it inspired in me a powerful tool for creative logical investigation, e.g., where any topic can be considered by examining limits and boundaries. The bouncing back and forth from one extreme to another drives an exploration towards the truth.
You did a great job of explaining basic calculus. I’m 75 and was introduced to calculus at age 15.
I wish you had been my maths teacher all those years ago, because I really understood for the first time! 🇬🇧
You really should explain the "dx" symbol. I was an outstanding math student in high school. Then I got to college and had to take calculus, and failed miserably. I tracked it back to this symbol. My lack of understanding of this symbol interfered with my abili8ty to do calculus. And I've never found a satisfactory explanation for it since.
Here's why it didn't make sense to you (or anyone else with a brain)...
Calculus Foundations:
Contradictory:
Newtonian Fluxional Calculus
dx/dt = lim(Δx/Δt) as Δt->0
This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale.
Non-Contradictory:
Leibnizian Infinitesimal Calculus
dx = ɛ, where ɛ is an infinitesimal
dx/dt = ɛ/dt
Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.
Well I guess the closest thing I can describe it to is like (1/(infinity))*x. It is an infinitely small of part of x, there are infinite of these parts. When you integrate you add all these infinite parts to get x. For anything higher than integration of dx I don’t know. I just solve without thinking about it since it isn’t in my syllabus. The best thing I can think of is when you integrate f(x)dx, you take a tiny part of f(x) and add it another infinity of itself(ignore this I don't really know).
I’m the same. I never found someone who could explain it to me. I suspected my maths teacher didn’t understand it either. It’s like floating rate notation. People use it and just accept they don’t understand it.
I meant floating point notation of course. Mixed my metaphors! 🙄
Omg Navy Nuke School has a 48% failure rate amongst applicants who can already DO 'at shit!@@MD-kv9zo
An exercise we did in one calculus class was to derive the formulae for various shapes (e.g. rectangles, triangles, circles, etc) using integrals.
I am an artist.
This post looks to be a good one for a non-math person.
I am getting a headache right now,
but I will revisit your post later when I have the time to focus my brain.
(I was a senior in high school in the junior algebra class.)
I am good with geometry, but, this a level above.
Thanks.
How is your headache? I hope it's better.
I like the way you prompt your discussion. It's so cozy and not intimidating at all. Like you are coaxing us all to sleep soundly with dreams full of numbers. And after I wake, everything in reality is great with numbers
Thanks, this lesson was just about the level where I dropped out of my communications electronics trade back in the 70's. Thing was some teachers/lecturers can teach and some simply can't. In my case it was simply me ! I think if I was just a little less burdened with life I could take up an interest, not sure how far I could go but maths has ALWAYS fascinated me. It's predicted so much, so many discoveries because maths showed the way. I enjoyed your lesson and at least now I will remember what that symbol is..integration, I hope.
I did learn in my pre degree ( 1978) ..felf more comfortable in solving differential calculus problems.. Thanks for your simple practical application of integral cals..presently working in an Engineering & Cont firm as Finance Manager at Muscat...
Great illustration of why so many struggle with math. Hey let's pull the number one out a hat and add it to the exponent with no accompanying explanation....etc.
cz u know it already.. hw abt those who never encounter this branch of mathematics.. be level headed bro
Man thank you. Maths is like a language, you either speak it or do sign language. Those who can speak maths remain the best in maths. And you sir are one the best teachers.
One think I thought was always lacking in my math education was that teachers would never explain what the application is for what was being taught...for many people, that would be very helpful in learning how the material could be applied. This video is a great example of this. There is zero mention of what this concept can be used for. If teachers did more of this, it could inspire many more people to be interested in math I think. what would be examples of x and y and how does that area under the curve telll us something.
The way I explain what integration is and its application is simply as accumulation. For example, if I accelerate at a certain rate, what speed will I be doing after 10 seconds? Accumulating acceleration gives you speed - the integral of acceleration is speed. What happens if I then accumulate the speed I got in the previous step, how far have I gone? Distance is the integral of speed. (And for derivatives, going the other way of course) Other applications, e.g. geometric, literally a use of measuring the area under a curve - imagine I'm designing a fuel tank that fits in an aircraft wing and has a constant cross section, which is curved. If I know what this curve is, I can turn it into a function, find the integral (the area under this curve), and work out the cross section and then the volume by multiplying that by how long the fuel tank is. I can easily use a little bit of algebra so I can have a specification eg. the customer wants a 100 litre tank, and its cross section will be this, how long will this tank need to be? Then there's things like PID controllers, where I stands for Integral and D stands for derivative, both from calculus - it mixes proportional, integral and derivative values to produce a control signal, e.g. to keep an oven at a constant temperature. The proportional is how far off are we right now, the integral is how much error we've accumulated, and the differential is the rate of change. These are some simple real world examples that should really be introduced because as you say, in a vacuum, young students especially will wonder what it's for!
@mikeearls126, as a senior in high school in 1969, I thought the same thing. My councilor advised me to take Physics along with Trig/Advanced Math(Calculus was contained in this course). Everything @74HC138 mentions above along with light, particles, waves, electricity and many other things were taught in the Physics course. All the math needed for all of this was taught in both classes. Sometimes we learned what we needed in the math class but other times, we didn't get that far yet so the Physics professor taught us what we needed. To me, the combination kept me more interested than I may have been.
Exactly. We were taught it from a strictly math viewpoint, with no idea of what we were actually doing. IN the exams we crashed with integration or differential calculations. We didn't understand the practical application, so it was just mumbo jumbo to us.
If the questions were framed about how much money etc etc, or how many apples you could steal from a 8 foot tall tree etc etc given what ever the circumstances were, we were all experts on that sort of calculation!!!
For some unknown reason, we all understood money and stealing apples from neighbouring orchards. And we looked like angels... butter wouldn't melt in our mouths.....
I agree. If I had been explained how the maths was used practically it would have been more interesting. I did a plumbing course a few years ago and suddenly the maths I had learnt at school became more interesting and I also began to understand why some formulas were as there were.
Thanks a lot for this amazing explanation. I did calculus I to III in campus 10 years ago but I survived by memorizing things instead of really understanding the concepts that you clearly explained here.
So no explanation of why you take those steps to find the area? Looks like magic to me!
4:00 You have to use the integration formula to solve for area problems with irregular shapes because integrals provide the best estimation for the area rather than just guessing and checking. So, rather than using a million little rectangles, the integral formula works to get the perfect area
BEING A MECHANICAL DESIGN ENGINEER, I STUDIED CALCULAS HOWEVER I NEVER USED IT & EVEN FORGOT ALMOST ALL OF IT, EVEN THOUGH I RELISHED IN ALGEBRA, TRIGONOMETRY & GEOMETRY WITH HIGH MARKS DURING THE 1960'S. BUT SEEING THE BASIC CONCEPTS ALL THESE YEARS LATER REFRESHED MY MEMORY. SO THANK YOU FOR YOUR VIDEO.
I loved calc in high school. Flow rates of change, position/velocity/acceleration, area/volume. Just working through orders of dimension was fascinating to me. Then I went to college, had a few lousy professors, and became a business major where division is considered complicated.
Good math profs are few and far between.
I'm 67, recently retired, and I failed calculus 3 seperate times after High School over about 15 years. The Calculus Book for Dummies was my fourth attempt around 2010. I did understand with following the book. But hand me a calculus problem and I'd never get it. Learning a few basic rules you covered here gave me a better understanding. And I have basic geometry like in your example to approximate the area under a curve. I understand fractions very well. In fact my high school chemistry teacher taught us 2 ways to do Stochiometry (balancing chemical equations) one with fractions, and the long way. Guess which one I went with.
They say "Information can never be lost." But....
Everything I learned about calculus is long gone.
The old saying goes " use it or lose it." Something complex like the language of math is if your not actively using it every day it slips back into the abyss !
It’s like Algebra and Logarithms, it’s not relevant to my day to day life, but in certain professions like computer science and engineering, it will be. The trouble is that we weren’t taught how mathematics are used in the real world away from the classroom.
It’s up to us to use these tools in everyday life. That is the value of education. It teaches you to think. Application to the real world was never going to be easy.
Most folks use basic algebra, but they don’t realize it.
That’s why true intelligence stands out. It translates seemingly meaningless information into tangible progress.
@@earlwoodland1873 Give me an example?
My biggest gripe is at the very beginning of the class No one explained why it was important to know how to do it that would have interested me more and I would have paid closer attention.
I have taught various classes once I was older and I always explain in detail what's the whole point of the class in the beginning.
It sets the right tone and I can refer back to what I told him the very first day as we go along so that by the end of the class they have the best chance of understanding the material and the reason to invest so much time and energy to understand what it means.
62 here . Back in university, I got straight A's in the complete course in single-variable calculus, but that's as far as I went, and because I never had to use it in real life (despite a 40-year career as a software engineer), the knowledge atrophied to where I couldn't do it to save my life. Thanks to channels like this, It's all coming back to me ... but I still haven't yet needed it in my daily life. 😕 Maybe I'll pick up a retirement hobby that requires it ... Any suggestions?
Where did the 1 in the first step come from? What was dx all about?
the dx is simpy abbreviation from differential-x
and yes the integration is 1-power up, and differentation is 1-power down. differential is slope or m;
example: dif. x² = 2x (the number 2 become multiplier/moved to the front, the power 2 reduced by 1),
if reversed, int. 2x = x², or int x = x²/2 (the power up +1, the number of power become divider)
hence int x² = 1/3 x³, int x³ = 1/4 x⁴ ... and so on...
that's what i can recall from my highschool in 80s 😂
I was an AMIE student in the early eighties preparing for Part I since I was a Diploma holder at that time.I vigourously prepared for Maths took tuition from one professor of Maths.Even though I could not succeed in my two attempts I still admire both Differential and Integral Calculus.I still have one book authored by Professor Hardy.
This was awesome. One question. How did you know to add 1 to the exponent? Is it because the difference between 2 and 3 is one? In other words, if we were going from 2 to 6, would you add 4?
I'd like to understand that too.
@JenningsB9 guess we'll never get an answer.
I will not bother proving why it is so but it is always "+ 1" (for functions similar to the one used in the example which f(x) = x^2). Going from 2 to 6 would not change that. The final answer would change though because it would be equal to this: 6^3/3 - 2^3/3 which is equal to 6x6x6/3 - 2x2x2/3 = 216/3 - 8/3 = 208/3 = 69 1/3. (The reason for the "+1" would be too long to explain... sorry!) By the way if the curve's formula is f(x) = x^5 instead of the example shown, the integral would again have a "+ 1". It would be x^6/6... being between 2 and 3 or whatever two values on the x-axis doesn't play any role in this. Hope this helps a little bit!
I think this answer could use Espo to bang the puck back to Henderson. The “+1” is part of what’s called “the Power Rule”, one of the many “tricks” (shortcuts) we use in calculus. @TeamCanada72 is correct, the proof is long and abstruse, but there are numerous explainer videos available on TH-cam and elsewhere
Could you mention one name of the explainer on yt please. @@BobDobbs68
Thank you for An excellent presentation. I am a retired engineering doctorate. This reminds me of my college times where we had to find the distribution of charge in a plane with a singularity represented by a pin hole knocked into the plane. I enjoyed returning back to basics. Hope the young generations have teachers that explain maths in such a simple way as you did.
Regarding 2:38, that figure is known as a right-handed side cutting lathe tool bit.
I'm 67 years old. This explanation was very clear and made me revisit the calculus I had almost forgotten I learned in middle school.
This inspires me to learn subjects I was always interested in but never thought I would understand. Great teacher!
Recall the original problem integrate X²dx with respect to x with limits set at 3 and 2. After integration, it will be x³/3 + C (constant). Substitute the limits 3³/3 + 3 - (2³/3 + 2) and so on. Loved this math and differential equations. It has been around 45 years since college.
pls make same video on basic introduction to algebra
It’s even using some algebra principals on how do we deal with fraction. It’s using every other math that we learned throughout the years and putting it all together.
Tnks , you made very easy .People who knows math will use your method
Back in my University days (1969 - 1973), there was no degree in Computer Science or even a department focused on computers - but wanting to get into compute programming, I could only go with a BS Math and take all the computer courses offered by the Math Dept.
I had to suffer through 16 semester hours of Calculus.
I survived it, barely. Got my degree, and then built a successful career as a Software Engineer, never having to do an integral or derivative, ever again. Thank God!
I have to add that I never had an instructor explain the very basic concepts of Calc like you have just done.
I wish I could say the same. I started my own Electronics engineering company totally weak on calculus and short-cutted and estimated my way though. I knew what the results should be so touched on it enough to get by but I wish I'd have seen this, it would have made my life a lot easier!
Thanks for sharing the knowledge
Thank you, I cant believe you did what no other teacher did. Explain what calculus does. I enjoyed it. accept for my question below.
Archimedes already developed infinite summation (integration) but unfortunately was killed by Roman rabble. It was rediscovered by Isaac Newton, 2000 years later.
Invented by two independent men who didn't know each other : Gottfried Leibniz and Isaac Newton. Today we use the Leibniz notation because it is more elegant.
Good point, i.e., Leibniz also invented calculus.
I think Archimedes was killed by a Roman soldier. Not sure.
Yes he was. The command was to capture him but the soldiers killed him. Otherwise we could be 1000 years more advanced.
@@paulanizan6159 interesting you should mention that long unnecessary gap in our intellectual history. For my history senior thesis in spring 1976 I wrote essay stating that there was 1800 years between Archimedes and Galileo in terms of the development of math. I blamed the Church mostly but my history prof taught medieval euro history and did not like my assertion. I stick by it till this day. The Church and Roman Empire squashed the logical progression of math for 1800 years. And I asked at end of essay "Where would we be today in progress of physics had a Galileo appeared in 100 BC?"
My experience with calculus is that most of its concept are quite simple, but quickly result in very complex algebra. There is also the logic of how to apply the rules you know to the problems at hand.
that big "S" means sum up...
from 2 to 3
integrate means to add a power.. i.e. x^2 becomes x^3
here... (1/3)x^3 + C
at x=3
(1/3)(3^3) + C
=9 + C
at x = 2
(1/3)(2^3) + C
= 8/3 + C
(9+C) - ((8/3)+C)
=9 - 8/3
=(27-8)/3
=19/3
= 6&(1/3)
That's right he forgot the constant entirely - tho those cancelled since it was a definite integral, still it should have been included as a general case
If I had a teacher as clear as this my life would have been quite different. Great video
It's been decades since I did that at school. I think it was non-parametric curves where it lost me. Interesting to revisit the topic so thank you.
Your explanation and introduction into calculus is brilliant!! If I had a math instructor like you back in high school and freshman college, I might have had an increased interest and done better. Of course, paying attention and asking for help could have worked too….😊. Just want to say thanks. Love TH-cam. Damn, I’m fifty years too late.
TabletClass Math, could you please explain how 'x squared dx' becomes 'x cubed divided by 3'?
Am I corrected that 'x squared dx' is the first derivative of 'x cubed divided by 3' ?
Thanks
A thought, does x**3 dx become x**4/4?
Just a story of my sophomore year at Aalto university (Espoo, Finland). It was a year of 1983 and I was doing serious math. One home assignment was to calculate of an area of 3-d object. It was defined with minimal info. It took me some 3 hours to make a mental picture what was the shape of the object. It was late Friday evening and my room mate was preparing to go out for fun. My fun was math. When I figured out that it was a banana defined with math formulas. Then it was easy to calculate the defined surface area. I was happy.
Oh dear! You skipped the most important point to understand.Yes, we are adding up the areas of the whole series of rectangular shapes. But each of the tiny rectangles has a height of x^2 and a tiny width of delta x, as delta x approaches zero. There is no understanding of this problem or any other without understanding dx represents the width of each rectangle and the concept of the limit as delta x approaches 0. Without this understanding it just becomes memorizing another formula.
It doesn't matter... as someone who really, really struggled with math in high school (and continues to do so), understanding this watered-down concept was a revelation ... I actually understood what what going on and my number-phobic brain stepped up a gear.
Thank you for this. I was wondering why he never explained the height part of this. Also it's not clear to me why he added 1 to the exponent. Why 1?
Absolutely right!Deriving the “formula” is the true beauty of integral calculus. Formulas, by themselves, are pretty boring.
When I was in school, none of this made sense for me and I was hesitant to ask questions. I am 43 now and trying to understand it. Thank you for the great explanation 👍🏻
It was my favorite math class in high school.
You're smart and you're way ahead of my generation I went to high school in the 1970s algebra itself wasn't even a mandatory class. You didn't have to take it😊
Hello Ralph, I went to high school in the early 70's. None of the 3 math courses were mandatory in my school.
@@paulanizan6159I hear you. The curriculum has gone up since our days
I cannot say this Ralph. I heard that university students can not even write cursive or do some simple math in their heads but need a calculator for the simplest calculation which I do in my head in a split second.
@@paulanizan6159I know somebody who has a degree in sociology, he needs a calculator just to do basic math he's terrible in math and he'd be the first one to admit it
When I was first learning calculus almost 50 years ago, my big stumbling block was trying to figure out why I should care about the area. I was more interested in the function(s) that bounded the area. Honestly, reckoning speed and acceleration was much clearer since I had real-world examples that could be applied much more easily than area. Either way, certainly at the beginning, I just applied the laws for integrals and derivates to get an answer without actually understanding the applicability of what I was doing. It was just math problem -> math answer. At some point it was more of an "a-ha!" moment than my teachers or professors explaining it so I really understood it. (I liked Jaime Escalante's, "I don't have to make calculus easy because it already is," from Stand and Deliver. And it kind of is...once you "get it".)
Calculus Foundations:
Contradictory:
Newtonian Fluxional Calculus
dx/dt = lim(Δx/Δt) as Δt->0
This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale.
Non-Contradictory:
Leibnizian Infinitesimal Calculus
dx = ɛ, where ɛ is an infinitesimal
dx/dt = ɛ/dt
Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.
Been 40 years since my math minor. This vid was a great reminder of the pain I suffered! lol. Enjoyed the refresher! Followed it perfectly.
Very cool! Is it always +1?
Thank you for giving me some start of an understanding of how it works. I missed a few days at secondary school and never caught up, but have always meant to look at it again.
I had forgotten even that y=x squared even. It’s almost like magic and I’m left wondering how on earth anyone could have come up with all this in the first place!
Great refresher, thanks.
Why would we need to learn this if AI can do the thinking
What a fantastic little film. I 'learned' calculus at 40 years ago in a very dry old fashioned way. So I could do the maths 'algorithmically' but never 'got it'. This was fantastic. A description of 'why' the rules are what the are would be more challenging to explain but would be appreciated.
Excellent explanation! Kudos!
Excellent 👌👌👌
I am a mathematics graduate and feel going to my mathematics nostalgia of my college days. Kudos Sir ❤🙏👌
I have always wanted to conquer my fear of math and learn more math than I did, including understanding the concepts behind the equations and formulas. Maybe this is beyond the basics, but I was disappointed when John didn’t explain why 1 was added to X squared. Without any explanation for why this was done, I felt left in the dark.
It was probably best as he did it. These are "the rules". You have the option to go on.
😢😢😢😢😢
I've been the university's top dog in calculus - differential and integral, it helped me understand where the formulas came from. And now after 20 years, seeing this video, I've completely forgotten about it, maybe because after college, I could not apply calculus in every day life.
But you don't say why x2 goes to x3/3. Is this not important? Is it just something we should just learn and accept rather than try to understand?
That’s what labanitz theorem derived
You made the explanation so easy, that each and everyone on the face of the earth can understand it! Your explanation is the real life example of George Polya's problem solving process should look like plus pedagogical talant! Congrats!
You just brought back the EUREKA feeling I had when I first learned this. Happy days 😊
I took 3 semesters of calculus in college, but that was 40 years ago. I also began to see it as a language to describe classical physics and engineering. Fun to review. Thanks. However I never used in my various careers.
Everything is clear until 12 minutes in. X cubed divided by 3. There's no explanation of why or where this comes from.
That's exactly what frustrates me. What's the reason behind that formula? Yes, it's easy to memorize this formula but curious minds must know how did the mathematicians conclude this to be the exact answer. I'll search a bit on that. Will get back to you if and once I find something of value.
Refer labanitz theorem from which it’s derived . Very basic is sequence and series, continuity of function and limits how can he explain everything .
He made the videos for how irregular areas are calculated . In a rough way
@jkj1459 he did explain everything up to that point and then, I felt, he said something with no explanation.
Thank you for opening up a world to me today. Despite a zillion ads that played throughout the video, I really feel like I learned
something very important today. Thank you.
An explanation that leaves too much unexplained. “Here we add 1.” Uh (raises hand) why do we add 1? Why do we add anything at all? Why not 5? Is 1 a personal friend of the instructor? Did 1 pay an endorsement fee? What’s wrong with 5? Has 5 been blacklisted? Is there some unseen prejudice at work? Is there no justice? I don’t think you should call it a “basic” lesson or introduction if you are going to gloss over a fundamental and integral part (ha, see what I did there?) or assume some specific insider knowledge on the part of the audience. IMHO
The title only says “basic” calculus. Which it is a basic calculus operation for teaching purposes. He also mentions he has more content available if you want to go deeper into any lesson. It’s good that you question, but even better is to be proactive on your pursuit of knowledge. Keep learning 🤙
I did the chatgpt thing:
“Adding 1 When Integrating: When you integrate (find the antiderivative), you add 1 to the exponent and then divide by the new exponent.”
🤦♂️
you need to go to the advanced lessons. This was specifically for the beginners.
I did Calculus at University for a while but never completely understood it. I think this has helped to explain something I probably didn't know or forgot. I would be interested to know why the integral works and relates to the rectangles.
I never wanted to take Calculus ; it was a required course when I went to college.
For you guys who don't understand the very first question of "Why would I ever need to know the area of under that curve?"
Many fields using these cartesian quadrant. From engineering, economics, coding, marketing, statistics, music, or even today's influencers. All of them need a tool to analyze stuff accurately. Calculus simply a way to operate one of the tools. Like, for example the Demand Supply curves in economics often need to find area under those. Which involving prices of products, or profits of that products, or decisions regarding that products, etc. Which ultimately affecting us all in the end of those.
Any Cartesian Quadrant will involve curves, then curves will involve calculus. Basically, everything involve calculus if you delve deep.
Scribble a random line in the quadrant and there will be a "name" for that scribbles (in this video, the curve's name is X^2). There is a way to find your random scribbles' "name", there is a way to find the area under that scribbles, and then there is a way to find *VOLUMES* when you start involving the next axis of the "Z" axis (add another elongated "S"). So basically, a line into 2D area, then area into 3D volume.
Then those academics that using triple fold of those elongated "S" of integrals already talking in the next dimensions of axises (whats the plural of axis?). We cannot see it, but we still can calculate it. Its like a blind cant see the sun, but can measure its heat.
All of that for a simple rule of "Just add the power by one and divide with that."
School back then were simply trying to give us the manuals to use this tools, in hope that we can use it too. Just like my daughter's play group trying to teach my daughter how to use toll called pencils, my uni tried to teach me how to use tool called calculus.
The problem I had with calculus was that my teacher failed to answer the most basic question. "Why am I learning this $hit? To what purpose?"
I couldve done well if I wasn't baked most of the time
90 years old.There was a 70 years ago I could have solved that problem but I don't think I really understood it. Your method seems to be better than whatever it was when I took calculus.
Repeat yourself much?
Turned off before halfway - boring repetition