Lec 16: Double integrals | MIT 18.02 Multivariable Calculus, Fall 2007

Lecture 16: Double integrals.
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[@akhileshnevatia995]

12 years later, still the best lecture :))

[@shakesbeer00]

The geometric approach to calculus is so helpful in understanding all sorts of concepts.

[@FarFromEquilibrium]

This professor gets more applause in class than any other Ive ever seen.

[@andresarpi1143]

What a wonderful professor! May God bless his soul.

[@Basigek]

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!

[@Dawatehaq786]

Lecture 3 Vector addition ➕ properties of vector addition kzbin.info/www/bejne/gombaaeCbKqieNU

[@mathsolution9002]

nice saying this is so help full to us

[@Ramix09]

34:33 dat dramatic close-up though xD

[@maimouna704010204050]

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!!!

[@samiulbasirtasin40]

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.)

[@hilariousharry1890]

The real mvp bud!

[@g1ntok147]

Really Enjoyed this lecture..helped me to understand double integration by perfect visualisation...thank you prof.

[@YoussefCherqaoui]

I find it cool how he slides in some french words without people noticing :)

[@Jkfgjfgjfkjg]

Which words?

[@eclipserealtyAZ]

after going to my regular lecture, i watch MIT's and think- why do i waste my time going to mine

[@user-sf9qr9fj3p]

I feel the same !

[@Yeahagreed]

Lol same

[@DeepPatel-ij3ih]

What A Wonderfull Lacture ! I got the Real feel of Double Integrals... :-)

[@user-ms9vn3ur5g]

Yes !! For Real !!!! :-))

[@hagenfarrell]

Professor Auroux is just too good!

[@dbeast2585]

I wish we had teachers like him in India.

[@Firstlast-er5qi]

d beast we have and sometimes even better ones.

[@proghostbusters1627]

@@Firstlast-er5qi only an Indian would say that xd

[@mostafizurrahman2694]

Oh.. my god.. the board and chalk are phenomenal..!

[@fukgovernment]

Thanks very much MIT and Prof. Auroux, you are the best math professor I have ever seen....

[@dreia2405]

I paused the video when he wrote pi/2 and solved the integral many times getting pi/8 😢, thinking I was wrong

[@seanchen2990]

me too

[@MrCavitysChessCorner]

Phenomenal video. Thanks Prof Auroux (o est-ce que je dois dire merci?) and MIT for putting this together.

[@ravil4]

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.

[@alexxbuzzy]

He’s crazy smart

[@m13m]

single integral - area double integral - volume

[@pabloastoreca8726]

Mohd Maqbool Alam you save the world my friend

[@mrkakotube]

single integral: length/ area double integral: area/volume

[@spa2581]

What's tripple integral then?

[@navs8603]

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?

[@coreconceptclasses7494]

@@spa2581 hypervolume

[@kwanhimshek]

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

[@virginialikesyou]

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??"

[@shanks1847]

Wow 9 years later I have similar feelings

[@miguelarbelaez1847]

@@shanks1847 The names are found at the start of each lecture. In this case, the teacher is Denis Auroux

[@kartik6110]

That cos 4 theta integral is very simple with reduction formulae. :) 2/3* ( 3/4 * 1/2 * pi/2 ) = pi/8.

[@stavros808]

greece says that this is an amazing lecture

[@kevinlogue2141]

nice handwriting

[@malexmartinez4007]

33:31 Minnie Mouse learns Calculus 2. Great lecture!

[@nadekang8198]

I wish I had a professor like him

[@LINGFERNANDO]

I love this series. Thank you professor. Thank you MIT.

[@uberll]

great videos, these explanations help so much.

[@dr.samazizi8542]

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_ji]

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 👌👌👌

[@footballCartoon91]

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

[@africanchina1]

Simply excellent!! The Internet can save the world

[@additeachxo]

this man is an amazing teacher.. i wish i had him!

[@amrkhaled29]

He has great presentation skills (Y)

[@alinapol]

excellent lecture

[@alpistein]

ugh, I hate trig substitutions

[@arjunbanerjee2067]

who doesn't :P

[@HilbertXVI]

Arjun Banerjee Me, they're amazing

[@PedroOliveira-ez2ni]

@@HilbertXVI they are beautiful

[@codingWorld709]

Thanks Sir

[@killerof21121989]

may ALLAH bless you people with the best of this world... THANK U!

[@nikitabhatter2201]

Awesome explanation of concept!

[@t4llst4r]

Whenever he wipes the board I get SOOO excited...

[@nncarchs2149]

Amazing! Hope my professor was like you.

[@rajanjireddyrekulapelly4031]

Excellent lecture

[@playman350]

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?

[@alpistein]

yes but this only the second course in real analysis, none of the students have seen that yet.

[@footballCartoon91]

@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.

[@footballCartoon91]

@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.

[@footballCartoon91]

@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.

[@footballCartoon91]

@29:09 you can also find the area by pi/4 or 0.7853

[@footballCartoon91]

@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.

[@footballCartoon91]

@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.

[@jfisher890]

This video is very helpful, i appreciate the help.

[@not_amanullah]

This is helpful ❤️🤍

[@not_amanullah]

Thanks ❤️🤍

[@ZaggerG]

that e^y stuff blew my mind O_o

[@obaidaal-hashmy8042]

cos^4 (theta) could be integrated by using Wallis formula its easier :)

[@H4H0hl]

Amazing!! Really helped me in my studies

[@sandeepgrover395]

amazing lecture. loved it!

[@yonatanable]

in determining the range of of the area, the function depends on x....

[@chatcharitz]

It could be better if the camera was still

[@micortes89]

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.

[@adinsz]

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

[@WahranRai]

In french : Une tête de premier de la classe !

[@jorgeribes2662]

Let's go with integrals

[@philyaofilms]

e-2 is correct. (-e+2e-0-2) = e-2 and the other calculations are correct

[@EmeraldSky33]

@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!

[@AniruddhaWadnerkar]

what clean boards!!!, love them :)

[@methusalah2]

he has blessed berkeley last semester *_*

[@LAnonHubbard]

Awesome lecture, thanks!

[@shihaowang152]

Great lecture! Thx

[@478241]

Love it!

[@gyanabrota]

Plz explain fubini's theorem proof

[@robertsguitar]

God bless the internet.

[@blueswede5985]

"baby example"

[@odinheim]

the double integral is pi / 8 = 0,392699082

[@doronel]

Final problem seems wrong to me... should it not be -e - 2 ? Let me know people!

[@wehtam]

13:05 that S looks perfect

[@kavoos1000]

simple and great

[@toplow9278]

UCSC professor spent 30 minutes on what this guy did in 5.😢

[@coreconceptclasses7494]

Use beta function to solve at 38:45

[@hskup]

Support OCW!

[@mdkabirhossain2857]

Lots of love

[@ManavKaushal]

you do have him

[@g1ntok147]

37:56 can use the wallis formula

[@johnliamten]

Are you talking about the x= sin(theta) sub? I think anyone who's passed first year calc knows to do that.

[@prash1693]

Awesome........Thanks MIT :)

[@lisinka3]

@ 46:45 why did he put a negative in front of ye^y?

[@mahmoudeldesokey147]

Wonderful 👌

[@Suppboio]

8:45 that girl eating

[@HectorLobato-yl5pb]

Por que o professor é o Peter do ei nerd?

[@jonpit4342]

If you have Mathematica it saves you the trouble from computing the nasty trigs. Yes it is pi/8

[@kritikmunot3604]

43:40,32:00

[@alexxander7770]

Who is watching this video in 2020 october.

[@dragooner4]

were they booing

[@LAnonHubbard]

I hope the Flying Spaghetti Monster blesses them too :)

[@ashwinnatraj97]

Why does the range of y change for different values of x in the xy plane

[@mousedorff453]

+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.

[@vatsaldp]

A stands for arya. Arya stark Arya stark of winterfelf

[@therealgigglebop]

@gellscream2009 True but the rest of us don't understand you. It's ok though, artists aren't understood either ;D

[@alphareeboot504]

but the rest the lecture is pretty good!!

[@alphareeboot504]

the interation is wrong cuz the answer is 3pi/16

[@Dineshkumar-xv4xz]

Mind=blown

[@LNDRDALAN]

The courses of MIT seems really easy, in my university Tec de momterrey, we see more difficult and deeper lectures.

[@canalf007]

Same here in Chile, in Universidad Técnica Federico Santa María we started with double integrals a lot more difficult that that

[@grandorottcod1]

Innovation is what makes the difference. Those universities have a lot of money to innovate.

[@jamesrockford2626]

"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.

[@stevensantos8975]

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.

[@kizinfino]

James Rockford Totally agree...

[@mrkakotube]

So.. line integrals over scalar valued functions are not covered in this course?? How dare u!!

[@adinsz]

yeah he is right , of course, sorry teacher.

[@agingerrail123]

I feel that Khan Academy's way of explaining double integral seems clearer to me

[@josephwheelerton]

both great!