I knew the formulas and how to solve questions in Calculus. But for the first time I am able to visualize it and truly understand it. Thank you, Prof. Strang.
And it's a shame few know it intuitively like this. Calculus is one of the most beautiful discoveries in human history, and it's not even that hard to understand and yet gives so much explanatory power to things you see around you.
This series has been great for reteaching myself the Calculus I have forgotten before the next college term starts. In my past classes, the professor could never explain what dy/dx meant, or why dx was found after the function behind the integration symbol. I finally fully understand the notation. Thank you.
This guy teaches difficult topics like no other. No wonder MIT has the greatest. The difference between him and my high school math teachers is astonishing. Thank you MIT and Dr Strang.
Decided to watch these as a "refresher" since it has been years since I used any calculus...He does a great job explaining it, and he has an enthusiastic personality, which made it seem a lot less dry. I wish I had instructors like this when I was an undergrad...
I actually understand why integrals work now thanks to this guy! if you want to understand why integration finds the area under a curve, watch this video.
I've taken calc courses at UMD, even gotten good grades. This explanation sheds light on the ONE thing I never got - WHY calc works. Wonderful lecture!
Prof. Strang u are just amazing .....Me a student of class 12th could for the first time visualize all of this so well .... I can't thank u enough....This was truly very interesting lecture and because of u it seemed so easy.....
Love his lectures, just read that he was a Rhodes scholar, very impressive. I hope to be 1/2 as good as whatever it is I do as he is with teaching and mathematics.
Very good explanation professor! I have studied this integral part of calculus in a different way... Now I got a good picture of where this all came from. Thanks professor and all MIT people
Good lecture and terrific idea on giving a big picture on the topic. We are sometimes sink too deep in the bolts and nuts of the areas and lost the overview and forget why we are there... Hope there will be more lectures...
i love this kind of teaching suddenly things are crystal claer and make sense from the beginning.... great math this is how the real father of math , thought about math. other make it too complex and in the end it doesnt make sense so the student have to memorize and not really understand.
Thank you Professor Strang and MIT. My summer college calculus taecher is giving us definitions, equations, and then practice problems. I had difficulty seeing the entirety of the integral idea.
I wish I had a teacher like him. Education in my country is not expensive like in the US but the quality is poor, we have to learn by heart without understanding and it is a torture for me (I'm not a sheep). Using the differences between the numbers in a series is key to better understanding Calculus, if you want to understand Taylor series, think about the differences of the differences of the differences and how they can help you find the next number in a series.
This is all so incredibly easy. Then again, not many schools have a Strang-caliber faculty member on hand to predigest the subject and turn it into a child’s game.
Is it really easy? You need to know something about coordinate systems, the zero, geometry basics; adding, multiplying, powers, brackets; using characters of different alfabets, their meanings in math, and basics of algebra (equations without numbers)...
@@ankeunruh7364 Yea?? most singlevariable calculus deals with the cartesian plane. it is so basic that a 5th grader could come up with it if they wanted a coordinate system. using characters of different alphabets? by singlevariable calculus, the only greek letters I used was pi and theta.. basics of algebra arent hard, and no they're not "equations without numbers" they just represent a certain function to numbers (in the case of an integral)
inquisitive871 -- delta y is the height and delta x is the run of the slope, which is the ratio that comprises the derivative. The definite integral is the area of the function between two points.
Dividing up the area under the curve with rectangles is the conventional teaching method, but judging from the comments, you'd think it was some miracle insight that no other teachers had thought of explaining...lol.
A much better approach is to show the ideas behind derivatives and integrals as separate concepts, THEN prove they are inverses of each other. Faster, and most straightforward, AND lays the ground work for other things like numerical methods.
I wish my teacher taught like this... My teacher uses power points and it's horrible. I have to see how the problem is solved without steps being skipped.
@themanisheab180237ce I went to a catholic high school in the W.I. When I took my G.C.E's I only did o-level math. I guess I should have taken a-level math instead: since the first time I encountered intergral calculus was at college.
if i have a problem set of two functions & i have to determine one from the other how do i know which is function 1 & which is function 2 which will be the derivative & which is the integral is there kind of rule to know that?
EXACTLY how it should be, the method is a million times more important than the result, and some results such as simple equations and basic math become memory recall anyway
Simon Wallis so what? i learned this when i was 15 as well,no need to be a showoff.I bet that if you learned it when you was 15 you learned it in a lower level of understanding.
For a million things. Look up optimization problems. For example if you wanted to manufacture a can for selling soda and wanted to find out the cheapest way to make it while it can still hold 12 ounces...you use calculus for that. To see how to maximize square area if you have a certain amount of fencing....i can go on forever.
I knew the formulas and how to solve questions in Calculus. But for the first time I am able to visualize it and truly understand it. Thank you, Prof. Strang.
And it's a shame few know it intuitively like this. Calculus is one of the most beautiful discoveries in human history, and it's not even that hard to understand and yet gives so much explanatory power to things you see around you.
This series has been great for reteaching myself the Calculus I have forgotten before the next college term starts. In my past classes, the professor could never explain what dy/dx meant, or why dx was found after the function behind the integration symbol. I finally fully understand the notation. Thank you.
This guy teaches difficult topics like no other. No wonder MIT has the greatest. The difference between him and my high school math teachers is astonishing. Thank you MIT and Dr Strang.
i can't express with words how grateful I am about this class! Thanks from Brazil!
Decided to watch these as a "refresher" since it has been years since I used any calculus...He does a great job explaining it, and he has an enthusiastic personality, which made it seem a lot less dry. I wish I had instructors like this when I was an undergrad...
Absolutely love everything this teacher has to offer. My abiding gratitude to all who made this possible for me to learn.
I love you, grandpa
+sophie strang :D
+sophie strang :) same here
Such a rockstar.
LOL STOP LYING ;0
Awww this is so lovely
I actually understand why integrals work now thanks to this guy! if you want to understand why integration finds the area under a curve, watch this video.
I've taken calc courses at UMD, even gotten good grades. This explanation sheds light on the ONE thing I never got - WHY calc works. Wonderful lecture!
Just magnificent ! It takes a true genius to explain something like that in a way you gain deeper insight.
IF ONLY I had seen this video in 1988, I might have gotten > a C in Calculus 101! Beautiful. The whole connection with area has always mystified me.
Thank you Professor Strang. I hope you read these comments. You are very much appreciated.
this man is a wonderful teacher. I love watching him on things I thought I already new. I wish there was an algebra 1 he did so I could show my son.
I got A's in both calculus 1 and calc 2 and yet I did not know the basic concepts behind the math until now. Public Texas Universities for you....
Don't you have theory exams?
@@TadasG258 understanding why the rules exist is different from just memmorizing the rules
@@Sol-gl3nl But you can't pass a theory exam if you just memorize the content...
Sol the
DR. Strang thank you for another solid lecture on integrals.
Prof. Strang u are just amazing .....Me a student of class 12th could for the first time visualize all of this so well .... I can't thank u enough....This was truly very interesting lecture and because of u it seemed so easy.....
Lol
Love his lectures, just read that he was a Rhodes scholar, very impressive. I hope to be 1/2 as good as whatever it is I do as he is with teaching and mathematics.
Prof strang is a god amongst men...
Thank you very much to our old math guru. We are very much fortunate to get your ideas.
I think I would have saved a lot of time and effort in my research, if Dr Strang was my lecturer in university.
Prof. Strang ... awesome 👏 will be indebted to you forever
Very good explanation professor! I have studied this integral part of calculus in a different way... Now I got a good picture of where this all came from. Thanks professor and all MIT people
Thank you internet! This is the best time to live if you are yearning for knowledge. Thank you Professor Strang and MIT.
exposition is outstanding! this is how math ought to be taught. Thank you to all who played part in the production of educational such as this one.
Big thank you Professor Strang and MIT OpenCourseWare, from Australia.
Good lecture and terrific idea on giving a big picture on the topic. We are sometimes sink too deep in the bolts and nuts of the areas and lost the overview and forget why we are there...
Hope there will be more lectures...
Thank you very much Professor!
We are all sheerly grateful for your work!
This lecture is soooo precious😀 Thank you professor and OCW!
i love this kind of teaching suddenly things are crystal claer and make sense from the beginning.... great math
this is how the real father of math , thought about math.
other make it too complex and in the end it doesnt make sense so the student have to memorize and not really understand.
I love your explanation on derivatives. It was a good refresher and a nice mind exercise. Thanks for your time and energy.
One day I will get into MIT and tell him how thankful I am in person. He is just amazing.
This explanation is a piece of art.
the best video about integral
Thank you MIT!!! Thank you Lord Foundation!!
you forgot to thank the prof., Gilbert Strang
Great presentation, never been better explain!
even i have learnt all of these things that prof delivers in this lecture, it's still an interesting lecture :) thanks sir
great job professor we in Brzil apreciate your hard work.
i liked seeing where ylast-yfirst comes from that was really insightful
Good Job Professor
Many thanks.... a genius, an artist of math...
Thanks for your really good work. We all need more...
Thank you Professor Strang and MIT. My summer college calculus taecher is giving us definitions, equations, and then practice problems. I had difficulty seeing the entirety of the integral idea.
Dr. Strang is a Wonder of Nature and the best teacher ever ! He simplifies the complex as no other person can.
You're absolutely brilliant.
Thx to MIT and the Proffessor for posting this.
I wish I had a teacher like him. Education in my country is not expensive like in the US but the quality is poor, we have to learn by heart without understanding and it is a torture for me (I'm not a sheep). Using the differences between the numbers in a series is key to better understanding Calculus, if you want to understand Taylor series, think about the differences of the differences of the differences and how they can help you find the next number in a series.
Very clear presentation, realy great!
here in Brazil, we learn it in college.
high school stops at derivades, and, most part of schools dont reach neither limits.
good professor. I wish there were UA-cam 40 years ago when I took differential eq. course
This is why MIT is different. You focus on understanding the idea rather than memorise and solve
Thank you so much MIT❤️
@HamiAlDiar
What he is saying in arabic is that we wish to benefit from these amaizing lectures in the Arab world especially
@MasterDemon25 the audience is high school students, so yeah, it should be for 16 year olds... check the description of the course
This is all so incredibly easy. Then again, not many schools have a Strang-caliber faculty member on hand to predigest the subject and turn it into a child’s game.
Is it really easy? You need to know something about coordinate systems, the zero, geometry basics; adding, multiplying, powers, brackets; using characters of different alfabets, their meanings in math, and basics of algebra (equations without numbers)...
@@ankeunruh7364 Yea?? most singlevariable calculus deals with the cartesian plane. it is so basic that a 5th grader could come up with it if they wanted a coordinate system. using characters of different alphabets? by singlevariable calculus, the only greek letters I used was pi and theta.. basics of algebra arent hard, and no they're not "equations without numbers" they just represent a certain function to numbers (in the case of an integral)
Holy shit. So basically, delta X is the base, and the height is the slope. The sum of these areas gives you the area in the aggregate.
inquisitive871 -- delta y is the height and delta x is the run of the slope, which is the ratio that comprises the derivative. The definite integral is the area of the function between two points.
Thank You Professor! Last time I saw this was in 2003!
Beautiful courses
Bah...get to 33:32 and I still don't see how to actually *perform* the limit. We jump right to guessing the derivative..
that's what really fucked my mind , 1 hour and in the end we are back at guessing the fucking derivative.
I have to say , i get it now.
Yes, you kind of guess and you show you're right by doing the derivative back to get the integral argument
I really like this professor.
Dividing up the area under the curve with rectangles is the conventional teaching method, but judging from the comments, you'd think it was some miracle insight that no other teachers had thought of explaining...lol.
Great lecture. 👍🏼
thank you professor Gilbert :)
@aramian21 indeed... the applications to social sciences are remarkable too...
A much better approach is to show the ideas behind derivatives and integrals as separate concepts, THEN prove they are inverses of each other. Faster, and most straightforward, AND lays the ground work for other things like numerical methods.
Brilliant lecture.
I wish my teacher taught like this... My teacher uses power points and it's horrible. I have to see how the problem is solved without steps being skipped.
Edi IKR?!,Like good god i wanna see you solve the Eq and go step by step
I do love MIT.
Amazing Explanation
He is so good to be true!
Sir Strang, you are a legend!
Thanks for the effort Sir!
I wish MIT had video lectures on more math topics.
Thank you so much professor!!!! great review
@themanisheab180237ce I went to a catholic high school in the W.I. When I took my G.C.E's I only did o-level math. I guess I should have taken a-level math instead: since the first time I encountered intergral calculus was at college.
@gerardrbain1972 gerardrbain the question is which high school did you go to lol, this is high school math, this is basic calculus!
if i have a problem set of two functions & i have to determine one from the other how do i know which is function 1 & which is function 2 which will be the derivative & which is the integral is there kind of rule to know that?
29:22 Someone actually applauded..
Brilliant! Thank you, professor =)
Very good! Thank you!
this guy is genius af
nee kaa güzel anlatmışınız hocam .
Very wonderful sir
@MasterDemon25 you learned intergral calculus in high school? Which country did you go to high school?
He said infinitesimal!
So, what?
Wow that was cool. Very clear. :-)
If Archimedes had found out the connection between the first and second functions, he would have yelled, "EUREKA!"
Great job!
thanks you sir! this discussion is very helpful. :D
Beautiful mind!!
EXACTLY how it should be, the method is a million times more important than the result, and some results such as simple equations and basic math become memory recall anyway
instrumen apa yang bapak mainkan? What's musical instrument did you played yesterday, Sir?
I envy Mr Strang, I wish I could understand what he is saying!!!
Thanks from Sri Lanka
Hiii
Dream name
lol. In the UK some years ago we did this when we were 15 years old !
Simon Wallis so what? i learned this when i was 15 as well,no need to be a showoff.I bet that if you learned it when you was 15 you learned it in a lower level of understanding.
Sir what is use of calculus in real life
For a million things. Look up optimization problems. For example if you wanted to manufacture a can for selling soda and wanted to find out the cheapest way to make it while it can still hold 12 ounces...you use calculus for that. To see how to maximize square area if you have a certain amount of fencing....i can go on forever.
@aramian21 algebra? :)
woooooooo MIT! my dream school O__________O
Thank you
Some check space pictureing area detopic defind communiative space minus integrate equal Big Picture.
Try comment.
Love you ❤️