Line integral example 1 | Line integrals and Green's theorem | Multivariable Calculus | Khan Academy

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Concrete example using a 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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[@sajidullah]

tell u the truth..i studied this in 1975 and am now 59...Now I understand the damn line integral.

[@sontapaa11jokulainen94]

Congratulations!

[@williamrubin914]

Me too

[@objectoriented3049]

Now you're 70!

[@dreia2405]

It's fantastic how you summarize a one hour lecture in 15 minutes

[@oximas-oe9vf]

lectures are sometimes unreasonably long

[@oximas-oe9vf]

lectures are sometimes unreasonably long

[@thehorizonchaser]

If only I watched these videos before I started cramming for finals... Learning is so much easier when you aren't stressed haha

[@neelmodi5791]

thats my plan too

[@Peter_1986]

"Bah, several months before the exam, there is no rush." 6 weeks later "Oh yeah, that exam, maybe I should start studying now." Reads the first half. "I should probably review this first half, so that I really understand it." Two weeks before the exam. "Oh shit, I had better hurry now, let me see, I have a total of 30 sections to read, maybe if I read 4 sections a day and then practice old exams... or is there a better way...? Gah! Next time I will start studying in time!" Waits too long next time as well.

[@battlewing221]

@@Peter_1986 i thought this was just me lol

[@battlewing221]

exactly !

[@bachabacha7048]

@@Peter_1986 I'm watching this before one week 😹

[@md65000]

The Khan academy drinking game: every time he says something about "not too rigorous" or "hand waving" you throw back a shot.

[@TheFarmanimalfriend]

With a good bottle of Scotch, I would rapidly become unable to drive.

[@ChrisSudlik]

Thank you so much, these videos are about to save my education. I was struggling with line and surface integrals, and some physics electromagnetic stuff, The past few videos were great, and I really get the stuff now. It is really a great thing you are doing for people.

[@user-jk4jj4nm1q]

.. So what is Green's theorem?

[@ShayJorge]

Lol he has another video on it, don't know why he put it in the title of this video

[@erikmcguire5486]

I wish my professors would teach like this!

[@banshee511]

A good grasp of English would be a good start for mine...

[@karan853]

at 10:53 , for small method we can multiply and divide by 2. so integral part will be 2sint*cost that is equal to sin2t.then integration of sin2t= -cos2t/2 .

[@nikhilnegi9446]

Lol, I was also thinking the same Bro in India we are taught so many small tricks to solve the problems

[@astrobruce7585]

Lol ya

[@AhmedMahmoud-tv9vw]

nice one.

[@griffisme4833]

Or you can just know how integrals work and figure out that sinxcosx came from sin^2(x) and work backward relatively fast

[@darkseid856]

Just solve it using gamma function . Even faster .

[@theludvigmaxis1]

Amazing job. I haven't even started calc 3 yet but I like to jump ahead. Very nicely done it managed to click and I have a very good understanding in line integrals now!

[@EmpyreanLightASMR]

I'm going to pretend you jumped ahead by 6 years and are going to take calc 3 next year.

[@theludvigmaxis1]

@@EmpyreanLightASMR im actually a PhD candidate now, I took calc 3 probably the same year I watched this video. I was in highschool at the time

[@HAWXLEADER]

tip: try to hold shift while drawing a line to make it straight! it should work on all drawing applications... you are a master of teaching!

[@LucifertheAtheist]

Thankyou ever so much sir. This whole KZbin teaching programme of yours is extremely benevolent. Thankyou.

[@NiiAryee]

Line integral delivered in simple logical math. Amazing!

[@fahadnazir8924]

This video traces the journey of a super crazy integral becoming a super easy integral :-)

[@Sherlock354]

It seems strange to me but even when I feel sleepy and I see his lectures I can understand even then!

[@AjayiPelumi]

Sir, you just earned my respect! I wish I saw this before my first exam T.T Nonetheless, I'm now looking forward to the next exam!

[@ShayJorge]

Once again, Sal was able to teach me in 10 minutes what my college professor failed to in 2 hours.

[@okktok]

Or maybe you just aren't paying enough attention

[@dreia2405]

thank very much, you are an amazing teacher

[@nikhilgarg2578]

ANYWAY.....SUPERB WAY OF EXPLAINING STUFF IN A SIMPLE AND LUCIS MANNER

[@sandracordoba6090]

Your teaching abilities are beyond normal humans. Muchas gracias!

[@13Septem13]

Thanks a lot. Great explanation and example. Your teaching skill is great.

[@AWacademy521]

Amazing tutorial. Thank you dear!

[@abdelrahimabdelazim6963]

بارك الله فيك God bless you

[@Seracinfinity]

You have a real gift for teaching. Very rare

[@Nikifuj908]

I love it when the subtitles say "UNINTELLIGIBLE" in all caps.

[@parag183]

this video explains how radon transform work in ct scan... thanks a ton..u made such a complicated thing as easy as it can be...

[@erickjian7025]

nice one Sal. I think there is a easier way for sint cost: d(sin(2t))/dt = 2sint cost

[@goon5544]

makes perfect sense. I don't really feel like seeing the advanced proof if I understand.

[@MisterTutor2010]

I used that method as well. sin(t) cos(t) can be integrated to -1/2(cos(t))^2 or 1/2(sin(t))^2. The answer comes out the same both ways.

[@13Septem13]

hanks a lot. Great explanation and example. Your teaching skill is great.

[@minhvu8893]

I like the software you used in the video very much, can you give me the name of it?

[@tis_i_sana]

What was the need to do u substitution at the end ? It could easily be done by inspection...

[@katiemcgatie]

I followed everything until the end - why didn't you replace u with sin(t) at the end of your u substitution and keep your bounds of integration in terms of t?

[@MultiCoolman125]

It's a definite integral, so we don't really need to do that. We just need to evaluate it at those limits since the function is the same whether it is in terms of u or t.

[@mmelkon0]

Thank you very much. That was very well explained.

[@priyashanjayakody3413]

You are an amazing teacher. Period.

[@tomasiscool]

@pmcmena1 no because he reordered the bounds to fit the u function; at the end of the video it wasn't "from t=0 to t=π/2", but rather "from u=0 to u=1".

[@anisurrahman5498]

Thank you very much. That are the great explanation. ❤️

[@tomasiscool]

When you plot x(t) and y(t) on the 3d graphic, the quarter circle should begin and end on the x dimension extremity and the y dimension extremity of the cube, as in the function z=xy those extremities represent x=1 and y=1.

[@Hiei104]

Thanku very much sir for this and so many other videos..u are a great teacher..

[@l1mmg0t]

you are the best I wish 60 years ago my teacher can do this

[@robertwilsoniii2048]

We just performed a line integral! F*ck yeah! :D

[@firozahmed8430]

best teacher ever! 🔥🐯

[@wediadi6788]

What a fantastic lecture.

[@sandipsaha2090]

How can we virtualized the line integral if f(x,y) is a 2d vector, example Electric field vector

[@vulicuba8079]

dS (differential arc)is fundamentally non linear in the picture but it being infinitesimally small, it can safely be assumed to be linear thus allowing its approximation by Pythagoras theorem. Is this point of view correct?

[@lawrencesibusiso741]

I LOVE YOU MAN!!!!!!!!!!!!(writing in a couple of hours!!!

[@suryavikramrizal3462]

thank you so much Sal...

[@aryamaansingh1894]

Sal....u just saved my exam which is 3hrs from now...thanks a bunch :)

[@firdushhussain7043]

awesome...visualization

[@davidash80]

don't you have to put the values of u back into the equation after integration before finding the definite integral?

[@aylora7033]

i promise mr khan i will donate when i graduate ! 🥺💖

[@hanabekele194]

Thanks so much sir

[@MrCalhoun556]

If anyone wants to visualize vector and potential fields I suggest you to go to falstadcom. There is a java applet on this page which can generate these fields.

[@sarvagyagupta1744]

I started watching this so late but correct if me I'm wrong. An application of this could be to calculate the surface area of mountains? We know the trajectory of a mountain and the width in each direction, so we can integrate the area under the curve wrt height at each point. Am I right?

[@levmyshkin5590]

Fantastic!

[@lemons107]

Beautiful!

[@LucifertheAtheist]

I learn a lot from it.:)

[@soumyajain6982]

I love you 3000 for so nice teaching

[@D_dusze]

Brooo, im not even done with highschool and this now seems easy af

[@antoniofrancaib]

you are amazing man

[@SanjayYadav-zb4lg]

Can't we use the dot product between the function and ds? It will make it more simpler, isn't it?

[@CliveReyes]

What do the colors in the 3-D plane stand for?

[@hackman6486]

Achmed that’s actually a good question. They don’t stand for anything, it’s just a very smart way to tell you, the viewer, the direction of the slop and help you visualise by looking at the colour gradient.

[@123amouri]

Your videos are amazing !

[@078matthijs]

I had to pause the video saying that this is amazing!!!

[@Kopfgeldjaeger6000]

You make wonderful curly brackets!

[@abdelrahmanrashwan9392]

could you recommend any book that I can practice solving more problems

[@Peter_1986]

You might wanna take a look at the book "Calculus: A Complete Course" by Adams. Also, the site "Paul's Online Math Notes" has lots of sections for Calculus 1/2/3, with example problems and everything, and the KZbin channel "Professor Leonard" is a fantastic channel for those exact courses, with tons of full lectures by an amazing teacher.

[@yatharthsawarkar3607]

Excellent 🎉

[@runiyobeen12]

pretty neat

[@jadeitewu]

You Sir, are awesome.

[@elbodi9894]

befor i finish the video thnx realy it helped

[@melindaliao4136]

I'm counting on your videos to get a 7 on calculus

[@jcnotnot8120]

If you switch the variables back to t’s after the substitution you get -1/2, why’s that?

[@TheLionelreal]

if i consider a non parameterized function....like say i simply have y=x^2 ......should i parameterize it??

[@ghtkssk01]

Wow...that's great

[@TheGta4you]

Thanks Man (Morocco)

[@yichizhang795]

Thanks!

[@ninosawbrzostowiecki1892]

you can also integrate sint*cost dt by converting it to 1/2*sin(2t) dt right? I did and got the same answer.

[@aeroscience9834]

Yes, that works too.

[@santoslittlehelper06]

+Ninosław Ciszewski You can use half-angles, but honestly, u-substitution is a lot easier.

[@sebastianpearson7817]

+santoslittlehelper06 if you knew the double/half angle formula it can sometimes save time in situations like this, it requires less working if you already knew the formulae

[@cagataysunal1130]

@@santoslittlehelper06 If you're doing calculus you should know your trigonometry.

[@rdevg]

amazing.......

[@dorol6375]

I solved it with the identity sin(x)cos(x)=sin(2x)÷2

[@HotPepperLala]

wow when I retire, I think I am going to join khanacademy

[@muneneantonym1085]

good work sir,, otherwise make the video more visible

[@bb4856]

Why not mention that the factor you replace ds with is actually the length of c'(t) ?

[@joshuaminnies1388]

Been doing line surface and volume integrals for 2 months now only to realize now that the line integral is the area of the surface about the line. Face palm?

[@TuningFreak23]

Why did he use substitution instead of the product rule for the integrations of sin(t) *cos(t)

[@thevegg3275]

Sal. It seems you plugged the equation for the curve into where I thought you said the equation for the ceiling should be. I thought you said the ceiling was x times y but you've made it the exact equation for the floor.What if you have a problem where the equation for the ceiling is completely different than the floor? In other words, the f(x,y) (ceiling) should only be x times y not cos t times sin of t. What am I missing here, please? You cannot define the ceiling as a function of t, only the floor curve.

[@billydagenham]

I hope when my teacher eventually sees these videos he realizes that he's a incompetent union protected lower and commits hari kari

[@DrummerRF]

Yes

[@sumeetbvarma]

Thank thank

[@Ditiro100]

what software or device is used to make this videos?

[@OneOfTheManyJason]

why don't we just use double integral? looks like it can give the same area too

[@liam5010]

can anyone tell me what software he used to trace the f(x;y) curve please?

[@lariosphotography4721]

I understand the concept, proof of the formula and the results. But my question is, how is this useful in real life applications?

[@computerchemistry221]

Do contour integration

[@Anteater23]

What I don't get is how you get a plane from a line. All those green and yellow rectangles on the plane where never even on the line to begin with. Please help someone.

[@ssayyahh]

Can you help me with this question? The value of the integral of the function g(x,y)=4 x^3+10y^4 along the straight line segment from the point (0,0) to the point (1,2) in the xy-plane is? My answer with your method is coming 33*(5^0.5). ds part is coming as 5^0.5. The correct answer is being shown as 33 only. Need help?????

[@fahrenheit2101]

4 years late, but did you ever find out? I just tried it and I get the same as you.

[@tahagargoum6247]

hi to everyone sir just I want to know how you got this limits and how can I find it in other examples ?thanks

[@somebody4061]

The limits for "u=1" and "u=0"? Look up his videos on u-substitution.

[@somebody4061]

Basically, sin(0) = 0, and sin(pi/2) = 1.

[@Skystrike70]

9:45 giant OH moment

[@yahyasaad9758]

"Ill go switch back to that color I dont like" LOL

[@boyuanchang8708]

Thank you

[@nikhilgarg2578]

ALL WE DID TO GET A FUCKING 1/2........THAT IS SOMETHING THAT PISSES ME OFF

[@AuroraNora3]

That's the best part tho

[@curiousminds301]

Why you dont resubstutute sint in place of u