Introduction to the line integral | Multivariable Calculus | Khan Academy

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Introduction to the Line Integral
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Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.
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[@ddsslse]

Khan the savior of university students

[@arsindmp9202]

im almost 11

[@yewtwo]

@@arsindmp9202 weird flex but okay

[@brianahn7919]

Still now lol

[@gerardogutierrez4911]

jesse cole wow you’re almost 11? Omg people don’t usually do this advanced level math at that age. Are you a genius? I think you’re a genius cause I never met anyone who did calculus at 11 you must be the smartest in your town. Can you solve the integral of cos(x)? I bet you can

[@woofle4830]

The integral of cos(x) = sin(x)... it’s basic... lmao If he’s learning line integrals, he’d know basic integrals

[@lonelywuffy]

why cant some math professors teach like this? i sit in a math class for 1.5 hours and scratch my head. i watch Sals video for twenty minutes and makes total sense.

[@iwtwb8]

I've been thinking about how cool it would be to design a study where students are randomly assigned to normal instruction in math or are just shown Khan academy and given access to tutors, etc for help with homework. I'm guessing the Khan academy students would fare SO much better on standardized tests.

[@brianahn7919]

Super agreed

[@simplyexplained875]

Maybe you learn better in this environment than a classroom one?

[@BatkoNashBandera774]

To be an orator is an art, to be didactic is a skill. To be didactic whilst speaking eloquently results most likely from the mastery of his field. I know we idlolise teaching, and teachers -but sometimes it's not the A team who are leading the classrooms. It's a problem of pragmatics, most of us are lucky if we have two or three inspirational professors/teachers.

[@fahadnazir8924]

Exactly same

[@inflamedwithrage92]

I am in my 2nd year at college studying advanced Mathematics.And this is the first time that I have actually understood the reason behind doing integral f(x)dx. Teachers usually associate Calculus with mugging up the formulas and doing tons of numericals,but nobody even bothers to teach the theory behind it all.And the sort of explaination that you give cannot be found in the books.Kudos to you.This is just phenomenal.

[@mariedrapalova7365]

He is soooo good. Stewart's Essential Calculus can't even compare to this 10 minutes of common sense explanation. Thank you!!

[@BatkoNashBandera774]

But in Stewart you have a few references that will likely be useful later and if using MATLAB can include other interactive material to learn/hone mathematics more.

[@amandeepsaha]

13 years later, still the best explaination video on line integral.

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Don’t know if it’s a good news or bad one. 😂 at least we can say math is quite firm as time passes

[@Yuki-wn9kj]

agreed!

[@isaac5990]

This video taught me in 20 minutes what my professor failed to teach me in more than 2 hours. Sal Khan is the best!

[@sabretoothstudios9142]

your example of finding the area of a weirdly shaped wall just made this instantly click for me before an exam, you're my hero right now

[@phylton]

Do you know that pleasure you get from understanding the nature of something and seeing its genius? This video fits this perfectly. Thank you, Khan Academy!

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I shouldn't be paying my university as much as what I do considering the fact that I'm always on Khan Academy

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In my personal experience, you are the only teacher that has never failed to provide an intuitive understanding of math! Thank you so much!

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I can’t fathom the magnitude of my appreciation and the depth of my gratitude for this.... there aren’t simply enough words to express whatever it is I feel.... thank you

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OMG... You genius... Not one of my professor could teach me like that.....

[@lilj135z]

I'm sitting in my computer lab at school watching this and I finally got it. My jaw dropped and I said "Oh my God." Needless to say, the person on duty for the labs was not very happy about my outburst and kicked me out

[@vd853]

this makes so much sense. when i first learn it, i was just given the formula, and had to visualize the whole thing for myself, which made it even more confusing.

[@BarneyTang]

thank you so much. I read the textbook over and over for a couple times, and I still could not understand the concepts. The video is so intuitive that it totally blows my mind.

[@sofiiadruchyna3719]

I have read explanations from 3 different books and many articles trying to understand the logic behind the line integral and none of them explained it that well as you did. This makes so much sense! Thank you!!!

[@turboduck2902]

I started studying engineering this year and all my teachers tend to be terrible at explaining stuff. This was perfectly explained, clear, slow enough but not boring. I love you.

[@ryoukboy1010]

if anyone was confused, I took the liberty of deriving this myself before watching the video. So we know that {ds = sqrt(dx^2+dy^2)} which Sal shows very well where he got that from. Now, our goal is to get this in terms of "t". The function f(x,y) is already in terms of "t" because f(x,y)=f(x(t),y(t)). So, how do we get "ds" in terms of "t"? (By in terms of "t" I just mean, how does it change with respect to "t"). What I did was simply ask, how does "ds" change which respect to "t". First, ds changes with respect to "dx" and "dy". And Finally we ask, how does dx and dy change with respect to "t". If you can tell, asking by how something changes with respect to another thing is the same as asking for the derivative with respect to that thing. So first we asked, how does ds change with respect to "t". Which is the same as the derivative {ds/dt = sqrt(dx^2+dy^2)}, all I did was put ds over dt to solve for the derivative. Now we see that ds changes with dx and dy. And we ask, how does dx and dy change with respect to "t". Which is the same as taking the derivative. {ds/dt = sqrt(dx/dt^2 + dy/dt^2)}. Now for the most confusing part, we can actually move the dt from the left side to the right side by multiplying by dt. Why can we do this you may ask? Well, derivatives often behave like fractions (you can do research so I don't drag this comment out too long). Now we get something interesting when we do this. {ds = sqrt(dx/dt^2 + dy/dt^2) dt}. This is the moment that it should click for you. Now we can just plug in ds inside of the integral for the thing that we just derived and viola! We have the integral in terms of dt. Read this comment a few times if you find it confusing. This is what made it click for me. At this stage in Calculus you should be comfortable with manipulating the infinitesimals, like "dx" and "dy" etc... Good luck!

[@yuzhewang6746]

registered to class and stay at home watching this may be a more joyful college experience

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MAN, I am in love with your magical voice, goes right through to my brain!

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If a your searching meaning for Crystal clear,unambigious,vivid ....etc.Then Sal lectures will be the example to that. I am too much obliged and gratefull to Khan academy and Sal for thier service to student like us.

[@biaaaa22]

Thanks so much Sal. You always reminds me why I chose to study math.

[@rahbarjamil6367]

I didn't understand this concept anywhere as good as here..Thanks a lot sir

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He draws so perfectly even on a computer, Respect.

[@356379783]

crystal clear

[@randomIy]

thank you. i usually have to teach myself calculus because i don't understand my professors so it feels nice to have someone teach me. thank you again.

[@PersianAcademy]

Do not worry, this is the future of education ... I think in 50 years universities are still going to be around, but mostly as research institutes and where you get some hands-on work done, things you cannot do behind your PC ... traditional theoretic teaching of concepts will be all online.

[@edoardogribaldo1058]

you're the man! Thank you very much sir, it was mind-blowing I finally understood linear integrals!

[@giustinoblake3686]

this is actualy the best explanation of curve integrals i'v ever seen...and i'v seen o lot of them. great didactical work.

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Cannot believe this is free! Thank you so much Khan Academy!

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There's so much fun in understanding math intuitively

[@Mine4Craftin]

9:45, you understand us, thats why u are the best :D

[@Peter_1986]

+Brecht Van de Heijning My math teacher in Linear Analysis is terrible, he always talks with a monotone voice and expresses absolutely no emotions whatsoever (not even annoyance), he is just a blank slate who talks almost as if he is talking loudly to himself. Salman Khan, on the other hand, is a great example of what a math teacher SHOULD be like - he is laidback, he undersands his "students", and he doesn't obsess over countless proofs and rigorous fluff all the time that makes no sense to beginners anyway, but always makes sure to start with examples fairly quickly.

[@DerUnbekannte]

wow.. after many many hours of listening to my professors repeatedly explaining this, I was still in a state of confusion, 19 minutes of listening to you and I can say with confidence I now understand how this works.. thank you so much!

[@paradox6647]

This is the best explanation of the line integral I’ve ever seen.

[@i_izi_gambit7395]

am viewing this in 2018,..and i must say,this is the clearest explanation av ever seen!!...so so so clear!!..thank you!!

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LOVE THIS GUY! LOVE HIM!

[@lcy6434]

Clear lots of thoughts after years esp improving idea to momentum integral with piezometer.

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So much easirer to understand i love how you use the correct termsbut explain everythign in natural language so we can all understand it

[@GabriellaElkaim]

I think I just understood line integrals for the first time. None of my professors ever explained it that well.

[@DesmondLumait-ix7sk]

❤This is more than worth all that I learnt from my teacher.

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It's official, You are the greatest man alive.

[@zavutee]

I bow to You master Khan. You handle this way way better than my university teachers

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Maths can only be interesting when it’s taught this way. Idk why KZbin is better than the classes offered schools and colleges.

[@jpijacob]

you are a 100% genius. you explain it in such a simple words while others bluffing.thank you so much .

[@raqel3036]

11 years later and this is the best explanation

[@hakeemnaa]

clear explanation i don't know why the teachers can not explain it like this there is nothing is complicated here at all

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We are the luckiest generation, as we have learnt from world's best teacher, from another continent, from our home.

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If i could press the like button so many times in this video, i would press infinity times. What a great explanation!!! I hecking love you.

[@Jekvan]

Holly tears...I think all of the lecturers around the globe want to keep the beauty of calculus to themselves,or they just get sadistic pleasure of seeing grown students literaly cry in their lectures. This stuff right here,this is the way it supposed to be tought.Infact,it is better to kick out all of them overpaid professors and just screen this masterpiece at classes instead. Thank you.Thank you thank you thank you thank you!!

[@misana77]

I have a calculus exam tomorrow and now I understand where this weird square root of derivatives comes from. Thanks from Russia!

[@moths8274]

second year physics degree. we have a nanotechnologist teaching the maths module. this is lifesaving, thank you

[@shane12021]

Amazing. Was completely confused until I watched that! Thank you.

[@weishien]

its nice to see many students sharing the same problem of not understanding certain mathematical concepts over at this channel.... i don't feel alone anymore T_T

[@phipgorman5677]

Thank you for making knowledge free. You are a gift. You made this and many other concepts so easy to conceptualize for me, and millions of others. I am not in college. I just like to learn. Thank you for teaching.

[@adedejiadedamola9843]

Big thanks to you Khan Academy,i really appreciate, to understand the basic of this concept is really a great addition to my knowledge.Thank you

[@NTMihaila]

Wow. This makes perfect sense. What an elegant culmination of some pretty basic principles. :D

[@abdul-kareem4429]

It's the end of 2018 almost, and this video has been very, very, very helpful to me. Thank you!

[@mrblahhh67]

wow just amazing how clear you get the point across

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you are a legend. give this man a medal

[@paulgiovanni1136]

I'm not a student currently. But I'm reliving my past as a physics major then I need to redo all the calculus and differential equations. No one ever explained this to me like this. I never understood it till today

[@norwayte]

This is the best introduction into line integrals I have seen or read. I really looking forward for all kinds of line integrals and their interpretations. And looking for surface integrals... and Green, Stokes, Gauss. You are literally saving lives.

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Are you even alive?

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Epiphany of math teaching! Period!

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how can u be sooooooooo awesome at teaching!!!!

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@Thomas White I CAN'T BELIEVE YOU'VE DONE THIS.

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This guy is awesome..!! this really helps. you will actually start enjoying math.

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beautifully illustrated, very helpful!

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These are the only videos I've ever seen with more than a hundred likes and ZERO dislikes. Impressive.

[@hanzabass]

This is mathematical delight for the layman. Thank you SO much

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Guys i am from South Africa, when is Mr Khan's birthday? Lets do something for him world wide on his birthday?

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+Gugu Ngwenya Represent.

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he's too good

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would you like us to start by sending you a prepaid gift card of a $100.00

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Solve 100 questions.

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I am in bro

[@ronak226]

best video on the internet for Line Integrals

[@pakhigondhia9372]

woww... u made it easy sir.. i was thinking i could never understand line integral, but u make me wrong.. thank u sooo much

[@harishli2020]

ur lecture is just awesome........plz keep uploading videos like this......thank u...

[@francisbaffour-awuahjunior3099]

You're the best, thanks for helping me visualize this.

[@RebekahParkhurst]

Crystal clear explanation of line integrals. Ty!

[@anthonymmeek]

Careful with your dt/dt comment at 15:00. Differentials are operators, which you probably know, but I think saying that you can "multiply them" around is not a good habit. Other than that, I think the video, concept, and examples are clear to understand the problem.

[@forestdeep8483]

How is it that somebody can explain to me better through videos of 10-14 mins in length than my professors?? How?? how??? He makes it so clear and why can't my professors do the same thing?? He is an electrical engineer but I learned fluid dynamics more from him than my so called Professor of Aerospace Engineering. I credit him more than my professors as far as learning the material goes.

[@mitoand9]

Super well explained, this is how professors should teach it..!

[@fahadnazir8924]

I want to tell one thing, this is awesome❤️

[@vitortavares5624]

you are the God of knowledge

[@elfujo6595]

Wow. Thanks. That made it so clear. Wish I could give this a million thumbs up.

[@Lexyvil]

About time someone explains this visually for once.

[@sushobhitjakhmolaAdmin]

Best teacher ever.

[@BobClemintime]

Thanks for the reply, though I understood what the bounds meant, I was confused on how to get the bounds. I figured out later that the bounds are often given in the problem, but as I suspected you may need to determine them from intersections.

[@omarhamada2932]

man, you are simply the best.

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Thank you for making this video. It helped me during quarantine.

[@ellenlinlin]

thank you so much!!

[@wille2511]

My mind is blown!! Thanks for simplifying this!!!

[@harjitkaur4090]

What a way of explanation! When I ask my teacher to explain like that, they are like, we have given the formula use it and do sums :(

[@DeathEaterLink]

Excellent! Very straightforward logic, great for getting that intuition.

[@Tiwari_57]

Masterpiece class for line integral.

[@salawesome]

Best explanation of a line integral ever.

[@300483rahul]

Great sal u must receive a noble prize am not sure if u did !!!

[@ganeshkendre300]

but theres o nobel prize i maths....coz nobels wife had an affair wid a mathematician..

[@stampcrab]

You misunderstood, Rahul wants Sal to receive a noble prize, not some mere Nobel prize. I'm all for it Rahul! :D

[@j444n]

hey thanks for the video, you made the expression of an curve integral very clear to me! i wonder where exactly mathematicians wouldn't agree? is it themultiplying by the differential dt/dt or what, and why they would not?because it makes so much sense to me now you showed it. unfortunately my professor in class would not do anything like that but some strange other things most students and i never get, (physics students btw. ;))

[@clayragsdale7703]

Anotha one. Absolutely killed it on this one Sal. Mic drop

[@halaanbar-ko1ez]

you're incredible , thankyou so much .

[@masouddayaghi5923]

You are wonderful man,you teach and explain unbelievably nice.You are amazing.

[@anhtuanvo93]

Very good explanation, thanks you so much

[@guigaingamkamei9654]

Thank u so much. I'm speechless

[@douchelip]

great example; 11:50 really tied everything together.

[@ecaepevolhturt]

Point a is where the line integral starts and point b is where the line integral finishes. I'm sure in the next vid, when he does an example, things will become crystal.

[@petertaylor4954]

Thank you Khan Academy for all you do. 🙏🏽