Lecture 16: Double integrals. View the complete course at: ocw.mit.edu/18-... License: Creative Commons BY-NC-SA More information at ocw.mit.edu/terms More courses at ocw.mit.edu
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@akhileshnevatia9953 жыл бұрын
12 years later, still the best lecture :))
@shakesbeer0012 жыл бұрын
The geometric approach to calculus is so helpful in understanding all sorts of concepts.
@FarFromEquilibrium15 жыл бұрын
This professor gets more applause in class than any other Ive ever seen.
@andresarpi11433 жыл бұрын
What a wonderful professor! May God bless his soul.
@Basigek13 жыл бұрын
who the hell can put the deslike button in this ? its a free lecture, you become sick and you can find what you lost here!! thank you MIT!
@Dawatehaq7864 жыл бұрын
Lecture 3 Vector addition ➕ properties of vector addition kzbin.info/www/bejne/gombaaeCbKqieNU
@mathsolution90022 жыл бұрын
nice saying this is so help full to us
@Ramix098 жыл бұрын
34:33 dat dramatic close-up though xD
@maimouna70401020405011 жыл бұрын
My head was hurting in class just a few minutes ago. I couldn't get it and some one was saying "Oh easy so easy:" Why couldn't I see it? Then I search the topic. This video is the best! I kept thinking that my prof didn't explain anything! Or maybe I need something like this.Thanks for the easy to understand explanation. You just made my day, May Allah bless the person filming the video and MIT. Thanks. You just saved me. I thought I might fail the exam. Thanks a lot!!!
@samiulbasirtasin404 жыл бұрын
Lecture 1: Dot Product Lecture 2: Determinants Lecture 3: Matrices Lecture 4: Square Systems Lecture 5: Parametric Equations Lecture 6: Kepler's Second Law Lecture 7: Exam Review (goes over practice exam 1a at 24 min 40 seconds) Lecture 8: Partial Derivatives Lecture 9: Max-Min and Least Squares Lecture 10: Second Derivative Test Lecture 11: Chain Rule Lecture 12: Gradient Lecture 13: Lagrange Multipliers Lecture 14: Non-Independent Variables Lecture 15: Partial Differential Equations Lecture 16: Double Integrals Lecture 17: Polar Coordinates Lecture 18: Change of Variables Lecture 19: Vector Fields Lecture 20: Path Independence Lecture 21: Gradient Fields Lecture 22: Green's Theorem Lecture 23: Flux Lecture 24: Simply Connected Regions Lecture 25: Triple Integrals Lecture 26: Spherical Coordinates Lecture 27: Vector Fields in 3D Lecture 28: Divergence Theorem Lecture 29: Divergence Theorem (cont.) Lecture 30: Line Integrals Lecture 31: Stokes' Theorem Lecture 32: Stokes' Theorem (cont.) Lecture 33: Maxwell's Equations Lecture 34: Final Review Lecture 35: Final Review (cont.)
@hilariousharry18904 жыл бұрын
The real mvp bud!
@g1ntok1473 жыл бұрын
Really Enjoyed this lecture..helped me to understand double integration by perfect visualisation...thank you prof.
@YoussefCherqaoui7 жыл бұрын
I find it cool how he slides in some french words without people noticing :)
@Jkfgjfgjfkjg7 жыл бұрын
Which words?
@eclipserealtyAZ11 жыл бұрын
after going to my regular lecture, i watch MIT's and think- why do i waste my time going to mine
@user-sf9qr9fj3p2 жыл бұрын
I feel the same !
@Yeahagreed Жыл бұрын
Lol same
@DeepPatel-ij3ih7 ай бұрын
What A Wonderfull Lacture ! I got the Real feel of Double Integrals... :-)
@user-ms9vn3ur5g7 ай бұрын
Yes !! For Real !!!! :-))
@hagenfarrell5 ай бұрын
Professor Auroux is just too good!
@dbeast25856 жыл бұрын
I wish we had teachers like him in India.
@Firstlast-er5qi6 жыл бұрын
d beast we have and sometimes even better ones.
@proghostbusters16274 жыл бұрын
@@Firstlast-er5qi only an Indian would say that xd
@mostafizurrahman26942 жыл бұрын
Oh.. my god.. the board and chalk are phenomenal..!
@fukgovernment11 жыл бұрын
Thanks very much MIT and Prof. Auroux, you are the best math professor I have ever seen....
@dreia24057 жыл бұрын
I paused the video when he wrote pi/2 and solved the integral many times getting pi/8 😢, thinking I was wrong
@seanchen29907 жыл бұрын
me too
@MrCavitysChessCorner12 жыл бұрын
Phenomenal video. Thanks Prof Auroux (o est-ce que je dois dire merci?) and MIT for putting this together.
@ravil415 жыл бұрын
Thanks a lot for the videos; they were really helpful in preparation for exams. Though, I would suggest the operator not to zoom in as much, because it makes it hard to follow the flow of the lecture.
@alexxbuzzy5 жыл бұрын
He’s crazy smart
@m13m7 жыл бұрын
single integral - area double integral - volume
@pabloastoreca87266 жыл бұрын
Mohd Maqbool Alam you save the world my friend
@mrkakotube6 жыл бұрын
single integral: length/ area double integral: area/volume
@spa25815 жыл бұрын
What's tripple integral then?
@navs86035 жыл бұрын
Great point. In contextual sense, you can say single integral gives area under the curve and double integral volume. However, if you do a single integral with a disk rotating around x-axis from a to b, you get the volume of the solid. Isn't the interpretation case by case?
@coreconceptclasses74943 жыл бұрын
@@spa2581 hypervolume
@kwanhimshek11 жыл бұрын
got another way of slightly less clumsy integrating to pi/8 cos^4(t)=cos^2(t) - cos^2(t) sin^2(t)=cos^2(t)-1/4 sin^2(2t) double angle formulae again, original integrand becomes 3/8+1/2 cos(2t) +1/8 cos(4t) happy integrating
@virginialikesyou13 жыл бұрын
Thank you MIT, and can you PLEASE post the names of these wonderful professors so we can thank them by name too?? I go to a small community college and our instructors are wonderful.... but.. not quite like this :) These videos make me feel smart again when I was starting to ask myself "what's wrong with my brain??"
@shanks18473 жыл бұрын
Wow 9 years later I have similar feelings
@miguelarbelaez18473 жыл бұрын
@@shanks1847 The names are found at the start of each lecture. In this case, the teacher is Denis Auroux
@kartik61103 жыл бұрын
That cos 4 theta integral is very simple with reduction formulae. :) 2/3* ( 3/4 * 1/2 * pi/2 ) = pi/8.
@stavros8088 жыл бұрын
greece says that this is an amazing lecture
@kevinlogue214110 жыл бұрын
nice handwriting
@malexmartinez40073 жыл бұрын
33:31 Minnie Mouse learns Calculus 2. Great lecture!
@nadekang81985 жыл бұрын
I wish I had a professor like him
@LINGFERNANDO13 жыл бұрын
I love this series. Thank you professor. Thank you MIT.
@uberll8 жыл бұрын
great videos, these explanations help so much.
@dr.samazizi85422 жыл бұрын
Changing the order of integration in double integral is indeed equivalent to using the integration By Parts. Let's see how for this example: Put $f(x)=\int_x^{\sqrt{x}}\frac{e^y}{y}dy.$ Then $f'(x)=\frac{e^{\sqrt{x}}}{2x}-\frac{e^x}{x}.$ and so $xf'(x)=\frac{e^{\sqrt{x}}}{2}-e^x.$ The required integral is $I=\int_0^1 f(x)dx.$ Apply By Parts by considering $u=f(x)$ and $dv=dx$ to get $I=(1f(1)-0f(0))-\int_0^1(\frac{e^{\sqrt{x}}}{2}-e^x)dx.$ To find the first integral, use the u-substitution $u=\sqrt{x}$ and the integration By Parts again!
@dubey_ji6 жыл бұрын
I loved this lecture and yeah I integrated cos^4x .. ...it felt like a live classroom it was not boring like other video lectures on KZbin it was lovely 👌👌👌
@footballCartoon912 ай бұрын
The area of a quarter of a circle is not pi/8 but pi/4. when the radius is 1. Area of quarter of a circle= pi*radius*radius/4 radius=1, Area=pi/4
@africanchina115 жыл бұрын
Simply excellent!! The Internet can save the world
@additeachxo11 жыл бұрын
this man is an amazing teacher.. i wish i had him!
@amrkhaled2911 жыл бұрын
He has great presentation skills (Y)
@alinapol11 жыл бұрын
excellent lecture
@alpistein9 жыл бұрын
ugh, I hate trig substitutions
@arjunbanerjee20676 жыл бұрын
who doesn't :P
@HilbertXVI6 жыл бұрын
Arjun Banerjee Me, they're amazing
@PedroOliveira-ez2ni3 жыл бұрын
@@HilbertXVI they are beautiful
@codingWorld7093 жыл бұрын
Thanks Sir
@killerof2112198912 жыл бұрын
may ALLAH bless you people with the best of this world... THANK U!
@nikitabhatter22016 жыл бұрын
Awesome explanation of concept!
@t4llst4r11 жыл бұрын
Whenever he wipes the board I get SOOO excited...
@nncarchs21498 жыл бұрын
Amazing! Hope my professor was like you.
@rajanjireddyrekulapelly40314 жыл бұрын
Excellent lecture
@playman35010 жыл бұрын
Instead of using the double angle formula and expanding the square he could've used complex exponentials to linearize cos theta to the fourth, right?
@alpistein9 жыл бұрын
yes but this only the second course in real analysis, none of the students have seen that yet.
@footballCartoon912 ай бұрын
@7:56, I do not really understood limit, when he says "take the limit as dAi is approaching 0", I assume that when adding more area into the calculation, do so in a slow manner. It could be with respect to x or y axis. Let say , when adding / stacking more area/region into the calculation do so by 0.00001 or something like this. Let say in this case you want to calculate the volume of a tower, so start from the base region, after that you calculate the area of the region at the same x value but at current_y=previous_y+dy. In which dy in this instance is near to 0. It could be 0.1, it could be 0.001, etc but not 0. In other words, you are hiking this tower only a little bit and calculate its area there. Then you continue until you reach the top.
@footballCartoon912 ай бұрын
@12:32, when he said that S(x) is an area of a region parallel to yz plane , that means only at certain value of x. This way you only get the area of the region when x is at some point. To get the whole volume, you must move x to the front by small amount, and at the same time, find what is the new value of S(x) when x is moved slightly to the front.
@footballCartoon912 ай бұрын
@15:52, so he says, that the range of y to be integrated is depending on x value itself. So, the ultimate goal is to find the volume, to do so we must kind of move S(x) which is area with respect to x , that is move from back to the front, whilst also moving to the right ie across y axis, but since the shape of the polygon is irregular at different x values, we may have different ranges of y that only need to be integrated, otherwise the area is overshoot. From the area drawn, it really look like a single line, but it is really an area when small changes of y and small changes of x occur at the same time.
@footballCartoon912 ай бұрын
@29:09 you can also find the area by pi/4 or 0.7853
@footballCartoon912 ай бұрын
@30:52, if i have the relationship of x and y like so: where y is also f(x), : f(x)=x**2+3x+2, does that mean the upper boundary of range of y is between 0 and x**2+3x+2, if at the same time i want to integrate from 0 to 1 with respect to dx for x axis.
@footballCartoon912 ай бұрын
@44:14 suddenly when integrating dy with respect to e**y/y in the range of x until sqrt(x), you get area under the graph with the line y=x becomes as the boundary, and the area under the graph does not fall below it.
@jfisher89011 жыл бұрын
This video is very helpful, i appreciate the help.
@not_amanullahАй бұрын
This is helpful ❤️🤍
@not_amanullahАй бұрын
Thanks ❤️🤍
@ZaggerG12 жыл бұрын
that e^y stuff blew my mind O_o
@obaidaal-hashmy80429 жыл бұрын
cos^4 (theta) could be integrated by using Wallis formula its easier :)
@H4H0hl11 жыл бұрын
Amazing!! Really helped me in my studies
@sandeepgrover3957 жыл бұрын
amazing lecture. loved it!
@yonatanable13 жыл бұрын
in determining the range of of the area, the function depends on x....
@chatcharitz10 жыл бұрын
It could be better if the camera was still
@micortes8915 жыл бұрын
Maybe a faster way to do that integral is by parts taking u= (cosx)^3 and dv=cosx, shouldn't be hard to get that 2/3 integral of cosine to the 4 is equal to 2 times integral of cosine square! in that interval then use double angle fomula. But that's not the point of the lecture.
@adinsz15 жыл бұрын
game over ! (Pi / 8)*4 =Pi /2 so our teacher says that a cercle'area of 1 for radius, is Pi/2, game over, the answer it's PI
@WahranRai2 жыл бұрын
In french : Une tête de premier de la classe !
@jorgeribes26624 жыл бұрын
Let's go with integrals
@philyaofilms14 жыл бұрын
e-2 is correct. (-e+2e-0-2) = e-2 and the other calculations are correct
@EmeraldSky3313 жыл бұрын
@virginialikesyou This is Denis Auroux. It says the names at the beginning of each lecture! You can search for him in the "people" search at mit.edu and shoot him an email - I doubt he'll respond, but he might read it!
@AniruddhaWadnerkar7 жыл бұрын
what clean boards!!!, love them :)
@methusalah213 жыл бұрын
he has blessed berkeley last semester *_*
@LAnonHubbard10 жыл бұрын
Awesome lecture, thanks!
@shihaowang1529 жыл бұрын
Great lecture! Thx
@47824111 жыл бұрын
Love it!
@gyanabrota6 жыл бұрын
Plz explain fubini's theorem proof
@robertsguitar14 жыл бұрын
God bless the internet.
@blueswede59853 жыл бұрын
"baby example"
@odinheim12 жыл бұрын
the double integral is pi / 8 = 0,392699082
@doronel15 жыл бұрын
Final problem seems wrong to me... should it not be -e - 2 ? Let me know people!
@wehtam3 жыл бұрын
13:05 that S looks perfect
@kavoos100014 жыл бұрын
simple and great
@toplow9278 Жыл бұрын
UCSC professor spent 30 minutes on what this guy did in 5.😢
@coreconceptclasses74945 жыл бұрын
Use beta function to solve at 38:45
@hskup14 жыл бұрын
Support OCW!
@mdkabirhossain28574 жыл бұрын
Lots of love
@ManavKaushal11 жыл бұрын
you do have him
@g1ntok1473 жыл бұрын
37:56 can use the wallis formula
@johnliamten11 жыл бұрын
Are you talking about the x= sin(theta) sub? I think anyone who's passed first year calc knows to do that.
@prash169313 жыл бұрын
Awesome........Thanks MIT :)
@lisinka314 жыл бұрын
@ 46:45 why did he put a negative in front of ye^y?
@mahmoudeldesokey1475 жыл бұрын
Wonderful 👌
@Suppboio8 жыл бұрын
8:45 that girl eating
@HectorLobato-yl5pb2 ай бұрын
Por que o professor é o Peter do ei nerd?
@jonpit43427 жыл бұрын
If you have Mathematica it saves you the trouble from computing the nasty trigs. Yes it is pi/8
@kritikmunot36045 ай бұрын
43:40,32:00
@alexxander77703 жыл бұрын
Who is watching this video in 2020 october.
@dragooner410 жыл бұрын
were they booing
@LAnonHubbard10 жыл бұрын
I hope the Flying Spaghetti Monster blesses them too :)
@ashwinnatraj978 жыл бұрын
Why does the range of y change for different values of x in the xy plane
@mousedorff4538 жыл бұрын
+Ashwin Natraj Because y is dependent on x. I assume you're talking about the limits of integration. They would clearly change because the region which the integration is being done over has changed.
@vatsaldp4 жыл бұрын
A stands for arya. Arya stark Arya stark of winterfelf
@therealgigglebop13 жыл бұрын
@gellscream2009 True but the rest of us don't understand you. It's ok though, artists aren't understood either ;D
@alphareeboot5044 жыл бұрын
but the rest the lecture is pretty good!!
@alphareeboot5044 жыл бұрын
the interation is wrong cuz the answer is 3pi/16
@Dineshkumar-xv4xz5 жыл бұрын
Mind=blown
@LNDRDALAN10 жыл бұрын
The courses of MIT seems really easy, in my university Tec de momterrey, we see more difficult and deeper lectures.
@canalf00710 жыл бұрын
Same here in Chile, in Universidad Técnica Federico Santa María we started with double integrals a lot more difficult that that
@grandorottcod110 жыл бұрын
Innovation is what makes the difference. Those universities have a lot of money to innovate.
@jamesrockford26269 жыл бұрын
"If you can't explain it simply, you don't understand it well enough." - Albert Einstein Perhaps your instructors aren't very good? or not very knowledgeable ? or don't really understand the topic? The K.I.S.S method is the key to teaching and subsequently learning.
@stevensantos89759 жыл бұрын
Alan Avila you realize this course is just for people who need to learn MVC for their degree or something, not for people going into math/physics majors. MIT has courses on multivariable calculus with theory.
@kizinfino9 жыл бұрын
James Rockford Totally agree...
@mrkakotube6 жыл бұрын
So.. line integrals over scalar valued functions are not covered in this course?? How dare u!!
@adinsz15 жыл бұрын
yeah he is right , of course, sorry teacher.
@agingerrail1238 жыл бұрын
I feel that Khan Academy's way of explaining double integral seems clearer to me