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!!
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! 😂
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.
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.
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.
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.
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.
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.
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.
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.
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
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!
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.
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
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.
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.
Many students would be saved difficulties, a sense of hopelessness, and a possibility of rejection of mathematics if the instructors would take the time to explain or provide information fundamental to the topic being taught. John has been doing an excellent job in this respect.
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 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.
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.
calculus is like trigonometria. Math Languages. Trigonometria is ofcource for triangles but the concept is that humans can build languages to solve problems of the their cosmos. Great presentation, You love what you do. Greetings from Athens Greece
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.
Thank you for ,this refreher course (for me) as I learned integral calculus in 1967 but so many years have gone by and I didn't use it during my like as an engineer in Electronics
I studied calculus in high school maths class and I loved it. But I haven't used it since, and your simple explanation took me back to the enjoyment I got from it! Thanks. :D
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".)
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
Do you have a video that includes the next level of depth for calculus, such as an deeper explanation of the rules, the construction of a calculus equation, and the "why" of the operations you described here? I think that is where most of us got lost in school, because it was a matter of, "do this," instead of, "here is WHY you Do This."
What a great video. The thing I really hated about my high school senior and 1 st year University calculus were the text books chosen for us which simplified nothing and the forced rote memorization of trig function differentials.
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.
Retired after 40 years as a Civil Engineer specializing in Construction Estimating. Never used Calculus once. Computer software such as Agtek did the work for us.
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.
I was good at Math but got off track with trigonometry and calculus. I gave up the idea of becoming an engineer and became a veterinarian. I love my work every day, but I always wondered what would have happened if I had gotten a good teacher for calculus.
If only the teacher had explain this at the start of the course I might not have dropped out. But this was back in the 50's, maybe they do a better job now. Good job.
Exactly...and also where did the 3 come from. As to the primary formula, why is the dx there if it ain't important? I understand what he seeking to achieve but in this instance I think he's short on explanation
The plus one is because you are using the power rule to try and find the antiderivative of the function x^2. So basically, x^2 is the derivative of (x^3)/3. Finding the antiderivative, in simple terms, is basically doing the inverse of finding the derivative. The derivative of x^2 is found by multiplying x by the power and reducing the power by one. While the anti derivative is adding 1 to the power and dividing by the new power and you need to add the constant, but the x^2 function had no constant to add. Look up power rule for antiderivative or integration, pretty sure there are better and clearer explanations out there.
Sorry small edit, i wrote "adding the constant" twice, i corrected and removed the first mention because when finding finding the derivative, the constant becomes 0 if there is one.
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.
Thanks for that explanation - it was helpful - However a question: 6.3 what? Units? What units? What was the area of the rectangle? What are the units of the x/y axis? Are they the same as the rectangle? When if the rectangle had an area of 1000 sq feet? Would you just ad 6.3 more feet, or? I'm still a little confused.
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
i 'm fine with maths always enjoyed it but have never done any calculus at all in any way , so i was intrigued to find out how you calculate a random odd shaped figure , I was impressed and with you all the way until the final it's 6.3 approx - whether it's my ocd or what lol but to me an approximate answer is no answer it's either correct or it is'nt I knew 6.3 couldn't be correct strainght away when the answer was to one decimal point when it's recurring. Maybe I am missing something but it seems like a best guess maths. But thank you for explaining it.
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?"
How did you derive the parabola equation? Visually the rectangle descends below the X axis on the blackboard, which it can. So is your solution inclusive of the X axis or only where the parabola transects the shape? Or do you automatically place the X axis at the bottom of the form? If you do that then you have to try to fit the parabola accurately in combination with the rest of the form???
Also, you use different nomenclature than I have seen elsewhere. I usually see it as f(x) = formula. It would simplify things if mathematicians would use the same symbols.
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?
Very good but! The area of square, rectangle, triangle can all be easily deduced. The area of a circle is, in fact, a calculus problem because it involves a curve. Whilst is is widely known solution it still is a ‘calculus’ problem. In the straight line areas they are quite finite in getting to the area. A circle on the the4 hand needs to summing of ever decreasing squares to ‘estimate’ the ultimate formula pi x rad(2). The calculus proved the ‘estimation’ to exactness. Now this may not be 100% accurate but it would be better if you included the circle as a case of a curve in calculus.
I was thinking about putting a swimming pool in the backyard. The pool would be very curvy, yeah a very strange shape, and 3m deep at the shallow end and 6m deep at the other end. I wondered how much concrete I would need and how much water it would take to fill it. Furthermore, how many pavers I would need, for the surrounding area, and how much mulch I would need for the surrounding gardens. I have no idea how to work all this out. Perhaps you would consider doing a video on it. 😁
this is great. only thing i felt was my brain kept wanting the dx definition immediately. I don't think it would take away from the understanding because the understanding is distance overtime area in the illustration. Why not just tell everyone that? if you leave it just as you say it, it doesn't require any additional explanation. In fact, you could probably just use the cursor to just show in the video distance over time of the area. Isn't that the whole point? The very first thing I always say...throw a baseball or view a plane......it's just going in an arc distance over time. Everything is. Rain water on a wing with friction, the wing itself, the plane, the air, hair, etc...therefore. That is Calculus definition.
Well at least I know what calculus is used for - I often wondered how the areas of weird shaped objects such as sails (ones with curved leeches) can be measured. I had concluded you could get an approximation by dividing it into small shapes that can be measured. But how does one work out the formula for the curve? Also there are some steps of the integration that not explained such as when you add 1 to the 2 (squared) number and then divide by 3 etc? How did anyone figure out that was the correct approach?
What did you use to display, computer, different colored drawing, etc? Very nice ... like to use to explain plumbing concepts. Could u do video on how used tools to explain math concepts? Great calculus explanation, now wonder why it sounded complex. Thank you for keeping it simple, cogent, and easy to understand.
hmmm.. it took me 53 years to understand this hahaha. And kids today is having a hard time learning. God if you tube is avalable 40 years ago it will be a blessing to us.
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.
As an architect I have always “seen” the geometry of shapes. Having taken a fair amount of calculus I understood the concept of visualizing the area. However, rectangles, squares and triangles are easy to see and calculate and our tools (back in the old days) were t square 45 and 30/60 triangle. I always thought if we had calculus at our finger tips we’d make much more exotic shapes. Voila, the computer arrives ready to do that for you and look at modern architecture. Calculus in the flesh.
The hardest part of Calculus is the translation from the real world into math. (Deciding what the formula is that you can do the math on.) My former students hated word problems because they'd have to figure out what formula to use. I assume that doing something like programming a CNC to produce something would be very much the same thing. How to mathematically describe what you want to make.
Calculus leads to some strange lines but I'm also interested in a perhaps simpler form of calculus that calculates the rate of change of volume over time.
In the first part of the solution; you added 1 to the exponent of x making it x cubed instead of x squared. Why did you add 1? Is it because of the difference between 3 and 2 is 1? So if the width was say 2 to 5 on the x axis instead of 2 to 3 then you would have added 3 to the exponent of x?
Intergral calculus (this equation) is the opposite of derivative calculus. the derivative of x squared is 2x... so the intergral of 2x is X squared... start with derivative caculus.
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
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
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! 😂
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.
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
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!
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.
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.
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
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.
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.
This inspires me to learn subjects I was always interested in but never thought I would understand. Great teacher!
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.
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.
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
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!
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.
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
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.
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.
Many students would be saved difficulties, a sense of hopelessness, and a possibility of rejection of mathematics if the instructors would take the time to explain or provide information fundamental to the topic being taught. John has been doing an excellent job in this respect.
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.
Great refresher, thanks.
Why would we need to learn this if AI can do the thinking
Tnks , you made very easy .People who knows math will use your method
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.
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.
An exercise we did in one calculus class was to derive the formulae for various shapes (e.g. rectangles, triangles, circles, etc) using integrals.
calculus is like trigonometria. Math Languages. Trigonometria is ofcource for triangles but the concept is that humans can build languages to solve problems of the their cosmos. Great presentation, You love what you do. Greetings from Athens Greece
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.
Thank you for ,this refreher course (for me) as I learned integral calculus in 1967 but so many years have gone by and I didn't use it during my like as an engineer in Electronics
This isn't so bad! Got Calc. 1 next semester and you're giving me some hope here. Thanks for sharing this!
They say "Information can never be lost." But....
Everything I learned about calculus is long gone.
Where were you 45 years ago? Best explanation. Never used it my profession so I lost it. Retired now, but still find math interesting.
Sounds exactly like me!
Nicely done; you have a profound way of conveying technical information.
I studied calculus in high school maths class and I loved it. But I haven't used it since, and your simple explanation took me back to the enjoyment I got from it! Thanks. :D
pls make same video on basic introduction to algebra
Regarding 2:38, that figure is known as a right-handed side cutting lathe tool bit.
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".)
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.
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
Do you have a video that includes the next level of depth for calculus, such as an deeper explanation of the rules, the construction of a calculus equation, and the "why" of the operations you described here? I think that is where most of us got lost in school, because it was a matter of, "do this," instead of, "here is WHY you Do This."
What a great video. The thing I really hated about my high school senior and 1 st year University calculus were the text books chosen for us which simplified nothing and the forced rote memorization of trig function differentials.
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
Although this makes absolutely no sense to me, Thankyou for taking the time to make this video. From the comments you've helped many people.
Retired after 40 years as a Civil Engineer specializing in Construction Estimating. Never used Calculus once. Computer software such as Agtek did the work for us.
Good to do it by brain power though.
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.
I was good at Math but got off track with trigonometry and calculus. I gave up the idea of becoming an engineer and became a veterinarian. I love my work every day, but I always wondered what would have happened if I had gotten a good teacher for calculus.
If only the teacher had explain this at the start of the course I might not have dropped out. But this was back in the 50's, maybe they do a better job now. Good job.
Nowadays, teaching Mathematics is quite easy and less boring because I use such clips to teach my students. They love the ICT bit.
You just brought back the EUREKA feeling I had when I first learned this. Happy days 😊
Very good basic intro. A brief explanation of why +1 would understanding? Tks
Exactly...and also where did the 3 come from. As to the primary formula, why is the dx there if it ain't important? I understand what he seeking to achieve but in this instance I think he's short on explanation
Agree so where does +1 come from?
The plus one is because you are using the power rule to try and find the antiderivative of the function x^2. So basically, x^2 is the derivative of (x^3)/3. Finding the antiderivative, in simple terms, is basically doing the inverse of finding the derivative. The derivative of x^2 is found by multiplying x by the power and reducing the power by one. While the anti derivative is adding 1 to the power and dividing by the new power and you need to add the constant, but the x^2 function had no constant to add. Look up power rule for antiderivative or integration, pretty sure there are better and clearer explanations out there.
@ragnarX4 thank you 😊
Sorry small edit, i wrote "adding the constant" twice, i corrected and removed the first mention because when finding finding the derivative, the constant becomes 0 if there is one.
You are a wonderful teacher😊
Excellent explanation! Kudos!
Thanks very much.
Very clearly explained
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.
I wish you were my math teacher growing up!
Thanks for that explanation - it was helpful - However a question: 6.3 what? Units? What units? What was the area of the rectangle? What are the units of the x/y axis? Are they the same as the rectangle? When if the rectangle had an area of 1000 sq feet? Would you just ad 6.3 more feet, or? I'm still a little confused.
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
Thank you. Love calculus.
i 'm fine with maths always enjoyed it but have never done any calculus at all in any way , so i was intrigued to find out how you calculate a random odd shaped figure , I was impressed and with you all the way until the final it's 6.3 approx - whether it's my ocd or what lol but to me an approximate answer is no answer it's either correct or it is'nt I knew 6.3 couldn't be correct strainght away when the answer was to one decimal point when it's recurring. Maybe I am missing something but it seems like a best guess maths. But thank you for explaining it.
Very cool! Is it always +1?
In love with calculus ❤❤my fellow Mathematician keep up receive my support I have subscribed
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?"
Excellent explanation
How did you derive the parabola equation? Visually the rectangle descends below the X axis on the blackboard, which it can. So is your solution inclusive of the X axis or only where the parabola transects the shape? Or do you automatically place the X axis at the bottom of the form? If you do that then you have to try to fit the parabola accurately in combination with the rest of the form???
Thank you, I wish I could have got this 26 years ago... Life would have been different 😂
Very well explained
Also, you use different nomenclature than I have seen elsewhere. I usually see it as f(x) = formula. It would simplify things if mathematicians would use the same symbols.
Why did you add 1 to the exponent on the X in the first place? What made you choose a 1 instead of a 2 or something else?
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?
HUGE HELP. thnak. YOU. SO much. !!!! GREAT JOB !!
Great introduction video!
Excellent video👍
Very good but! The area of square, rectangle, triangle can all be easily deduced. The area of a circle is, in fact, a calculus problem because it involves a curve. Whilst is is widely known solution it still is a ‘calculus’ problem. In the straight line areas they are quite finite in getting to the area. A circle on the the4 hand needs to summing of ever decreasing squares to ‘estimate’ the ultimate formula pi x rad(2). The calculus proved the ‘estimation’ to exactness. Now this may not be 100% accurate but it would be better if you included the circle as a case of a curve in calculus.
I was very good in Maths, especially Calculus , did Graduation, but its applications are not much known that time
I was thinking about putting a swimming pool in the backyard. The pool would be very curvy, yeah a very strange shape, and 3m deep at the shallow end and 6m deep at the other end. I wondered how much concrete I would need and how much water it would take to fill it. Furthermore, how many pavers I would need, for the surrounding area, and how much mulch I would need for the surrounding gardens. I have no idea how to work all this out. Perhaps you would consider doing a video on it. 😁
What is the explanation for dx?
this is great. only thing i felt was my brain kept wanting the dx definition immediately. I don't think it would take away from the understanding because the understanding is distance overtime area in the illustration. Why not just tell everyone that? if you leave it just as you say it, it doesn't require any additional explanation. In fact, you could probably just use the cursor to just show in the video distance over time of the area. Isn't that the whole point? The very first thing I always say...throw a baseball or view a plane......it's just going in an arc distance over time. Everything is. Rain water on a wing with friction, the wing itself, the plane, the air, hair, etc...therefore. That is Calculus definition.
Well at least I know what calculus is used for - I often wondered how the areas of weird shaped objects such as sails (ones with curved leeches) can be measured. I had concluded you could get an approximation by dividing it into small shapes that can be measured. But how does one work out the formula for the curve? Also there are some steps of the integration that not explained such as when you add 1 to the 2 (squared) number and then divide by 3 etc? How did anyone figure out that was the correct approach?
What did you use to display, computer, different colored drawing, etc?
Very nice ... like to use to explain plumbing concepts.
Could u do video on how used tools to explain math concepts?
Great calculus explanation, now wonder why it sounded complex.
Thank you for keeping it simple, cogent, and easy to understand.
Starting to make sense?
Thank you!
hmmm.. it took me 53 years to understand this hahaha. And kids today is having a hard time learning. God if you tube is avalable 40 years ago it will be a blessing to us.
dx is the infinitesimal width of the rectangles we are summing up (integrating).
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.
As an architect I have always “seen” the geometry of shapes. Having taken a fair amount of calculus I understood the concept of visualizing the area. However, rectangles, squares and triangles are easy to see and calculate and our tools (back in the old days) were t square 45 and 30/60 triangle. I always thought if we had calculus at our finger tips we’d make much more exotic shapes. Voila, the computer arrives ready to do that for you and look at modern architecture. Calculus in the flesh.
The hardest part of Calculus is the translation from the real world into math. (Deciding what the formula is that you can do the math on.) My former students hated word problems because they'd have to figure out what formula to use. I assume that doing something like programming a CNC to produce something would be very much the same thing. How to mathematically describe what you want to make.
Why did you add 1 to the x2’s to make them x 3’s?
It's been a long time, but doesn't your formula for a triangle only work for a right angle triangle? Thanks.
He said "you can easily understand this." And I believe him. 🎉
I love calculus. Mathematics is my best subject.
Where did the 1 in the first step come from? What was dx all about?
I know nothing of math, but I'll continue to watch your videos always for simplicity and understanding on the go. +1Sub.
Calculus leads to some strange lines but I'm also interested in a perhaps simpler form of calculus that calculates the rate of change of volume over time.
If you wanna figure out the area of a strange shaped park how would you find out if y= x2 or y=x3???
In the first part of the solution; you added 1 to the exponent of x making it x cubed instead of x squared. Why did you add 1? Is it because of the difference between 3 and 2 is 1? So if the width was say 2 to 5 on the x axis instead of 2 to 3 then you would have added 3 to the exponent of x?
Well explained !!!
What about the constant created after integrating?
An excellent lecture
where did the extra "1" come from that makes x2 (squared), become x3 (cubed)? Is that a calculus rule?
Intergral calculus (this equation) is the opposite of derivative calculus.
the derivative of x squared is 2x... so the intergral of 2x is X squared...
start with derivative caculus.