Double integrals and Polar integrals: Explained with 3D visualizations

[@EugeneKhutoryansky]

To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available). --To change subtitle appearance: Scroll to the top of the language selection window and click "options." In the options window you can, for example, choose a different font color and background color, and set the "background opacity" to 100% to help make the subtitles more readable. --To turn the subtitles "on" or "off" altogether: Click the "CC" button under the video. --If you believe that the translation in the subtitles can be improved, please send me an email.

[@Heisenberg1969]

ok

[@qna8485]

Very nice explanation.. Best 👍👍

[@ankitaaarya]

@Valentin Baker nice market strategy

[@dr.z101]

Add Arabic

[@abderrahmanemihoub8484]

That is what we need in math education, animation!

[@petarugrina9627]

Abderrahmane Mihoub And classical music ;)

[@bsharpmajorscale]

If you like animation, you should check out 3B1B as well!

[@3227998]

exactly! It makes understanding so much easier!

[@Justin-wd2vy]

Hard to do with a black board and chalk. No matter how smart.

[@spiderjerusalem4009]

*in any sort of education, to be precise. Even biology would have to be most dependable on that

[@maxlolo1411]

This channel is absolutely a scientific treasure! Très Bien

[@TheNaz01]

Truly one of the best intuitive examples iv ever seen

[@EugeneKhutoryansky]

Thanks for the compliment.

[@EugeneKhutoryansky]

If you like this video, you can help more people find it in their KZbin search engine by clicking the like button, and writing a comment. Thanks.

[@sayanmondal4570]

+TheExaltedPheonix Finding Volume , mass if Density given ,Finding Centre of Gravity of complex shapes etc...

[@mandeepubhi4744]

Please post a video on Cauchy Sequences,(Infinite Series)

[@EugeneKhutoryansky]

Mandeep, I will add that topic to my list of topic for future videos.

[@mandeepubhi4744]

Ok Thanks👍

[@Adi13234]

Eugene! Thanks so much for these videos, everything makes more sense now. Would you be interested in describing how fluids work (fluid mechanics)? Thanks again!

[@parjohansson3118]

Again, a very elegant visualization of a fundamental concept in mathematics. These videos should most definitely be used in junior high! A picture is worth a thousand words and an animation like this is worth a thousand pictures.

[@EugeneKhutoryansky]

+Pär Johansson, thanks for the compliment.

[@flynn9500]

13:22 *physics intensifies*

[@spacejunk2186]

You write bold by putting an * before and after the segment you want to write as bold. *Like this And this*

[@Complexitivity]

@@spacejunk2186 *like this?* EDIT: *oh wow...*

[@yousefabdelgaber7498]

*let me try*

[@Test-ri2kr]

Yousef Abdelgaber -LOL-

[@GottfriedLeibnizYT]

magnificent. that's gonna make it easier for multivariable calculus students to understand.

[@EugeneKhutoryansky]

+Gottfried Leibniz, thanks.

[@icarus313]

Pretty cool when one of the founders of calculus itself comes back from the dead to agree with your video, eh? Haha! ;)

[@Imanol_Fdez_De_Arroyabe_Zabala]

@@icarus313 lmao

[@TheBrainn]

still waiting for Isaac Newton to reply to this

[@EugeneKhutoryansky]

You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link: kzbin.info_video?ref=share&v=GHBMiscPE-g You will then be able to add translations for all the subtitles. You will also be able to provide a translation for the title of the video. Please remember to hit the submit button for both the title and for the subtitles, as they are submitted separately. Details about adding translations is available at support.google.com/youtube/answer/6054623?hl=en Thanks.

[@李康南]

Physics Videos by Eugene Khutoryansky Hello，I love your video very much. I'm from China, and I highly recommend you to upload your video to bilibili, bilibili has a huge number of users in China. I suppose your video will be very popular at there. 3BLUE1BROWN has done that.

[@李康南]

KZbin is good，but most of us are still used to bilibili. Really hope you to come to bilibili. ^_^

[@zf164]

The greatest minds find elegant ways like your own where science meets the arts. A company like Pixar or Dream Works should create a division of animation with you in charge.

[@abdulrahmann.9024]

Are you kidding?

[@Mattvieir]

I wish i had this video back when I was studying this topic! Excellent video and excellent quality! The rock music at the end was super cool

[@EugeneKhutoryansky]

+Matheus Monteiro, thanks. Glad you liked it.

[@VlanimationTales]

What a satisfying explanation of double integrals! It has opened the door to a brand-new world of calculus that I can't wait to explore further. 😁 (The classical music was also a nice touch!)

[@EugeneKhutoryansky]

I am glad you liked my video. Thanks.

[@Kazekage121]

I'm subscribed to all the major educational channels (i'm sure you know of whom i speak) and i can honestly say that they can ALL learn something from you. You deliver absolute quality videos from the top down. Your work is AMAZING! It's only a matter of time before you hit the 1M sub mark (and above), just keep doing what you're doing. - Loyal Subscriber

[@EugeneKhutoryansky]

+Warren Wilson, thanks for that really great compliment.

[@Mysoi123]

What a way of teaching! beautiful, clear, and simple. The animations are so lovely, they make me happy just looking at the colors. You also carefully break down the crucial processes slowly, allowing people to take their time to understand what is going on.

[@EugeneKhutoryansky]

Thanks for the compliments.

[@irvingdelgado6426]

I wish I had found this before, it's very clear.

[@MichaelHarrisIreland]

Thanks, you make me approach mathematics without fear.

[@yanggao2102]

Mozart: Serenade in G, K.525 in case anyone wonders. Excellent choice of music!

[@michaelwang1730]

You may as well just call it Eine Kleine Nachtmusic

[@davidwright8432]

Extremely clear explanation and videos! Thanks too for keeping the commentary pace slow, with pauses so you can actually THINK about what's being said!

[@EugeneKhutoryansky]

Thanks for the compliment about my explanations.

[@hazemmmm9105]

you know what, a person makes clear and understandable videos like this should definitely understand the subject matter he explains, therefore our school teachers never understood the practical reflection but only have they known how to calculate.

[@dheibeljr]

This was very helpful for someone that is very visually oriented. THANK YOU!

[@EugeneKhutoryansky]

I am glad my video was helpful. Thanks.

[@phy-geek892]

An absolutely beautiful amalgamation of maths&music...with insane animations making the topic much simpler to understand!!!...

[@EugeneKhutoryansky]

Thanks.

[@diegocuevas8092]

SO MUCH CLARITY!! I love your style of teaching, it should be the new norm. Thank you so much for the knowledge. I think i speak for everyone that watches this when i say: It is much appreciated!

[@EugeneKhutoryansky]

+Cuey IsDope, thanks. It is nice to be appreciated.

[@davidb5205]

13:22 That Naruto music. Hahaha. Great video as always, I really wish I had these when I was back in college taking Calc classes. I'm a visual learner and seeing things makes them 'click' in my head.

[@EugeneKhutoryansky]

+David Boucard, thanks for the compliment. Sorry I didn't have these videos ready back when you were in college.

[@aerosmith4119]

One of the most beauty product for mankind. Big love, big thanks

[@EugeneKhutoryansky]

Thanks!

[@hc590]

Amazing!!!! One of the best videos explaining double integrals and the coordinates systems

[@EugeneKhutoryansky]

Thanks for the compliment about my video.

[@jayedhasan4835]

Best Channel to learn about in-depth calculus. It helped me visualize the calculus.

[@EugeneKhutoryansky]

Thanks. Glad to hear that my videos are helpful.

[@jayedhasan4835]

What software did you use to make those? In Bangladesh 99.99% student can't even imagine why we do double integration, or even integration. I'm a game developer myself. I wish I could teach those through games. haha

[@EugeneKhutoryansky]

I make my 3D animations with "Poser." I make some of the 3D models with "Wings3D."

[@dante1054]

Really interesting as usual I love these kind of videos

[@EugeneKhutoryansky]

+Dante T, thanks. I am glad that you like my videos.

[@seanwiesen7418]

Another amazingly intuitive explanation, Eugene. Thank you!

[@EugeneKhutoryansky]

+Sean Wiesen, thanks.

[@kca698]

Great classical example -in both calculus and music :) Great job E.K. !

[@EugeneKhutoryansky]

Thanks!

[@LoLMaaN312]

This is apsolutely brilliant, thank you for sharing these videos for free! Keep up the good work!

[@EugeneKhutoryansky]

Thanks.

[@fakherhalim]

Stunningly beautiful! Used to think physics and math are driest, now they appear prettiest! In less 15 minutes, I learned to calculate the size/area/volume of anything in the world! without any stress!It is Awesome!!!

[@EugeneKhutoryansky]

+Fakher Halim, thanks.

[@cool9930]

I cannot explain how much I love your videos

[@EugeneKhutoryansky]

I am glad to hear that. Thanks.

[@akshitkohli1806]

Best channel acc to a electronics engg. In india.. ! From india with love pls upload more videos with deep information.

[@EugeneKhutoryansky]

Thanks for the compliment. More videos are on their way.

[@Karmanya101]

I see that you too are a visual person, and these videos are so much fun to watch... I wish, I had these when I was learning. I do not mean just this video in particular ..but all of your videos. Good work! please keep it up.

[@EugeneKhutoryansky]

Thanks. I am glad you like my videos.

[@pepegasadge2977]

I was just listening to Mozart and when it was done I closed spotify and clicked on this video and then there were more Mozart!

[@ariclenescorreia6048]

Now I only understand integral and derivatives because of this channel. Congratulations

[@jacobslinares1983]

Best Multivariable Calculus Video ever!!! 🇺🇸🤘🏼👊🏼🎤🖖🏼

[@EugeneKhutoryansky]

Thanks for the compliment.

[@shrikantdeshmukh9052]

Awesome explanation. I never understood the concepts better before seeing these videos

[@EugeneKhutoryansky]

I am glad you liked my explanation. Thanks.

[@shrikantdeshmukh9052]

@@EugeneKhutoryansky Can you please also make a video on Navier Stokes equation please

[@johnmurray7767]

Please...surface integrals and differential equations.... Super explanations and graphics - thanks!

[@file2hd6026]

You are doing Gods work! Could you please start a computer science series?

[@EugeneKhutoryansky]

+Log Arithms, thanks for the compliment. Regarding computer science, I already have one video titled "Logic Gates from Transistors." Thanks.

[@DayaCIDfan]

the music was sooo dramatic. But anyhow, outstanding explanation. I just understood what exactly the polar coordinates mean. Thank you sooo much. I have never been told about the polar coordinates so well before this. Thank you very much.

[@EugeneKhutoryansky]

Glad you liked my explanation of polar coordinates.

[@TheShivajiFromIndia]

great service to the humanity. god bless you and your team

[@EugeneKhutoryansky]

+shivaji b, thanks.

[@henriknyholm6114]

I never comment, but i must say; this is high quality education for free and I love it!

[@EugeneKhutoryansky]

+Henrik Nyholm, thanks. I am glad you liked it.

[@howmathematicianscreatemat9226]

"The only really valuable thing is intuition" [Albert Einstein]. You not only understood this yourself, it seems like it's your signature. You are WAAAAAY ahead of most text books and almost all university lectures in terms of understandability. Because you actually present it in ways the developers thought about it in their minds. Truely remarkable mindsdont just start with the abstract definitions out of nothing, because nobody can truely think abstractly without having examples to abstract from. So you really do it the way the geniuses invented it but by also filtering out the most understandable parts. Hope you are now confident that you are a true treasure to most students out there.

[@EugeneKhutoryansky]

Thanks for the compliment.

[@Armc31416]

I remember from the Calculus classes that in the Polar integration the "R" multiplier was called the "Jacobian", so to differentiate from the Rectangular integration explained before. Amazing video!

[@polytopproducts880]

You just earned yourself another subscriber for making double integral so feasible. Keep it up ☺️

[@EugeneKhutoryansky]

I am glad to have you as a subscriber. Thanks.

[@Pri-tf9iv]

this is terrific stuff. I'm an aerospace engineering student and your videos about curl and divergence has really helped. Can i ask for a little help? concepts of CIRCULATION and VORTICITY is really confusing and probably you're the only teacher who can explain what they really mean. please do a video on above topics. thank you so much for all these amazing videos. :-)

[@EugeneKhutoryansky]

+Arpita Singh, thanks for the compliment, and I am glad to hear that my videos have been helpful. I will add circulation and vorticity to my list of topics for future videos. Thanks for the suggestion.

[@garvitgoyal5236]

Thank you so much for making such videos. The time you give between explaining two statement is really very helpfull.

[@EugeneKhutoryansky]

Thanks.

[@narendk6021]

madam u done great job.. i am in strggling to get understand about voltage and current for the past 10years.. atlast u make it easy ... thanks alot

[@EugeneKhutoryansky]

Glad you found my videos helpful. Thanks.

[@howmathematicianscreatemat9226]

Some say Eugene is a man. I don't know. But I think like you, that the Autor is a SHE.

[@janardankoiri5270]

What u find is actually double integration of dxdydz . It won't be double integration of zdxdy. But this channel helps me to visualize so many things. Thanks to your team.

[@markusherrmann9681]

Wow, never seen such a beautiful explanation of integrals. Very good work, thank you.

[@EugeneKhutoryansky]

+Markus Herrmann, thanks for the compliment. Glad you liked it.

[@dannycrofts8138]

Now I can calculate the volume of my entire rocky cake 🎂🍰😍👍🏽

[@janu5698]

great explanation....background music is soothing for the soul...nice

[@EugeneKhutoryansky]

Thanks. I am glad you liked my explanation and choice of music.

[@atibhu]

Great ! My all concepts are now cleared . Thanks .

[@EugeneKhutoryansky]

Glad my video was helpful. Thanks.

[@social.2184]

This channel is underrated.

[@EugeneKhutoryansky]

Thanks.

[@materiasacra]

At 5:47 I expected a third way: slicing horizontally in steps Dz. You then need to 'collect' the sets in the xy-plane corresponding to a particular slice. This gives the Lebesgue integral. It would have been so nice to see Lebesgue integration as an intuitive extension of the Riemann integration in x and y, instead of the usual 'arcane and formal' introduction at a much later stage.

[@amomisninos]

Well done explanation and visualization with beautiful music at first.

[@EugeneKhutoryansky]

Thanks for the compliment.

[@damianmussi3205]

One of the best videos I've ever seen... (I recommend use vel 1.25 or 1.5)

[@monkeypoohonyou]

This is the best video on integration I have ever seen! Just found your channel and have already subscribed! Excellent work!

[@EugeneKhutoryansky]

+monkeypoohonyou, thanks for the compliment, and I am happy to have you as a subscriber.

[@landricojr.garcia6505]

Very clear and understandable ever, even in grade school

[@EugeneKhutoryansky]

Thanks.

[@kazhiinisusana44]

Beautiful.........now its stuck in my brain....love it

[@EugeneKhutoryansky]

Glad you liked my video. Thanks.

[@YusefRhymer]

This was AMAZING and the music that came on at the end wow just wow

[@EugeneKhutoryansky]

+YusefRhymer, glad you liked my video and my choice of music. Thanks.

[@YashPrajapati-qi8xn]

You are my biggest crush Eugene.

[@cameronspalding9792]

This makes it so easy to understand

[@EugeneKhutoryansky]

Thanks. Glad to hear that.

[@feynstein1004]

Great work as always, Eugene. You need more recognition.

[@EugeneKhutoryansky]

+Feynstein 100, thanks.

[@BoIoko]

This is so intuitive. I wish I had this some years ago.

[@EugeneKhutoryansky]

+BoIoko, thanks.

[@sakinarajpoot7577]

No words that can express my feeling infact no words able to explain the goodness of this vidio i mean i dont know what should i hv to day legend thnkx for your work its soooooooo helpfulll

[@EugeneKhutoryansky]

Thanks. I am glad that you liked my video that much and that it was helpful.

[@sakinarajpoot7577]

@@EugeneKhutoryansky i m.became your fan its not norml.that u replied me oo my God m feeling soo happy aa worldbest teacher replies on my comment thankx alot sirr

[@jeffbezos3942]

The best channel on yt!

[@EugeneKhutoryansky]

Thanks for the compliment. I am glad you like my videos.

[@jeffbezos3942]

@@EugeneKhutoryansky You deserve much more popularity

[@EugeneKhutoryansky]

Thanks.

[@jhb8466]

I have the greatest respect for you. Thank you very very very very much. I really love your videos.

[@LinuxLuddite]

Another method to find out the area of circle is take those slices, separate into two groups and attach them in such way as to form a rectangle. the height of that rectangle would be the radius (R) and the length would be half the circumference of the circle. Circumference of a circle of is 2πr so half of it would be πr. Now applying the formula for area of a rectangle i.e. height*length we get r * πr = πr².

[@yousefabdelgaber7498]

Song title is morning mood by Edvard Grieg in case you were curious

[@tokajileo5928]

thi is great, just one note: at the end you end up with the area equation of the circle but you started to calculate the volume of a disk. in fact you calculated the volume of the unit height cilynder. could be confusing to someone.

[@wheelsdotfly]

OMG, i felt really exhited when I finally understood where the equation for the area of a circle is coming from

[@dellpi3911]

kzbin.info/www/bejne/f5LRmIt3eKuKapo

[@عَدِيُّ-م3ح]

he actually calculated the volume for a cylinder that has a height equal to 1 unit, but yes, you can also calculate the area of a circle the same way, it's super simple! :D. integrate rdtheta from 0 to pi/2 then integrate that times dr from 0 to R where R is the radius of the circle. and finally multiply by 4 since we only got the area of the first quarter of the circle. and you'll also end up with πR^2. But in the video the integration was meant for volume, not area. m^2 is not equal to m^3.

[@alfonshomac]

instant like!. I'm glad to see your subscriber base is growing.

[@EugeneKhutoryansky]

Thanks.

[@coldsoup49]

Great animations! I don't like the voice thing as much, but that's a pretty minor complaint. Overall this is some excellent content!

[@ЮджифПознань]

I really love your videos. It's make me understand math.

[@EugeneKhutoryansky]

Thanks. I am glad my videos are helpful and that you like them.

[@cameronspalding9792]

Volume=sum of arclength*height= sum of height*r*d theta*dr I never thought of it that way! I only thought of it because of area = integral 1/2*r^2 d theta and the Jacobian

[@multimrhp3008]

This channel is amazing.

[@EugeneKhutoryansky]

+MultiMrHP, thanks.

[@Electromowls]

You are doing a great job sir by making these videos👌🏻🙏🏻

[@EugeneKhutoryansky]

+Mouli Thirumalasetty, thanks. I am glad that you like them.

[@VTeslaV]

+Mouli Thirumalasetty Sir? How do you know it's not a woman? It's a female voice..

[@parjohansson3118]

+Nikola Tesla Kira Vincent--Davis www.imdb.com/name/nm1113042/ I had no idea she actually has a normal voice :-)

[@jhonramirez2659]

very good explanation, a video of the triple integrals explanation, please

[@EugeneKhutoryansky]

Thanks for the compliment. A triple integral is just a logical extension of the concepts shown here.

[@idrissberchil25]

Thank you so much, now it all makes sense where in class i just kept staring.

[@EugeneKhutoryansky]

Glad my video was helpful.

[@nitishkumarthaty810]

Outstanding teaching 🥇🥇🥇🥇.Thank you mam

[@EugeneKhutoryansky]

Thanks for the compliment. I am glad you liked my video.

[@funguyscienceman]

I've learnt so much from this video. Thank you!

[@EugeneKhutoryansky]

+Daniel Bostic (Funguyscienceman), glad to hear you found my video helpful. Thanks.

[@robbiemurchison7845]

This channel is the best

[@EugeneKhutoryansky]

Thanks for the compliment.

[@ull893]

Thank you. You are doing an amazing job in helping people like me who were always curious how we arrive at these weird formulas. Now I get it. Incredible amount of work you have put in. Thanks again. By the way, do you dream mathematics in your REM sleep ? Your videos are like dreams of revelations, like God or an angel taking me to the deepest knowledge libraries of this universe and showing to me, by holding my hand and explaining. Later I find I woke up with more wisdom, than I have gone to sleep with.

[@EugeneKhutoryansky]

Thanks for the compliment about my videos. I am glad to hear that they are helpful. No, I don't dream in mathematics. Thanks.

[@mahmoudeldesokey147]

Wonderful work And beautiful mind

[@tactics7127]

Respected Eugene, each rectangle has a different height along the z-axis. While you have considered that constant, what did I observe during the video that was without change in z values? Should it not have been calculated as a different value? While, at the end, it resulted in another integral, a whole volume would become a triple integral. Nonetheless, your videos are my motivation and encourage my confidence in physics.

[@reverberation_9]

Awesome videos, Eugene. Thanks a lot for sharing.

[@EugeneKhutoryansky]

+Oscar Dean, you are welcome and thanks.

[@HakaTech]

nice dude right before my midterms

[@EugeneKhutoryansky]

+HakaTech, Thanks. Glad to hear that the video was in time for your midterms.

[@pronoybhat]

Sir Its the best thing on You tube. Thanks

[@EugeneKhutoryansky]

Thanks for the compliment.

[@howmathematicianscreatemat9226]

I think it's a SHE :-)

[@paul88146]

what software do u use to make these? looks great, and very smooth

[@EugeneKhutoryansky]

I make my 3D animations with poser.

[@yousefabdelgaber7498]

@@EugeneKhutoryansky Why not blender? It's free and easy to use with plenty more features?

[@tiuk23]

I'm really impressed by your videos. I will promote them as much as I'm able to. They are the best so far I can tell. However, one thing is still out from my imagination - why the speed of light is unbreakable? If there is no such thing as "static frame of refecence", what is actually preventing me from reaching that speed, and how I can distinguish that I'm not moving faster than light in a certain frame of reference?

[@htheh7728]

Thank you so much,now I understand integration.

[@EugeneKhutoryansky]

Glad my video was helpful. Thanks.

[@huihui666]

Nice video! Would it be possible to make a video on visualising triple integrals? I understand this may be difficult, if not impossible, due to the added additional dimension. Also, can you also make a video showing the visual difference between scalar integrals and vector integrals? I can do vector integrals, but have a difficult time visualising what's going behind the scenes. Thank you!!

[@EugeneKhutoryansky]

+xihai luo, vector integrals are shown in my video titled "divergence and curl." As for triple integrals, all you have to do is extend the concept of double integrals to one more dimension. Or, if you want to stick with three dimensions, imagine that the three dimensional object is composed of cubes, and each cube has a number inside of it. We then add all the numbers inside all the cubes together. For double integrals, all the cubes would have the number 1 in them, but for triple integrals, the numbers can be any value.

[@ShaeljaMishra]

I just love your videos.

[@EugeneKhutoryansky]

+Shaelja Mishra, thanks. I am glad that you like them.

[@rb9025]

i love this video.. thanks a lot god bless you.. please do more videos and try to make every abstract concepts easy like that

[@adamdelawski5520]

Excellent !

[@Kamnuma]

The best video I have ever seen!!! Thank you a lot!

[@EugeneKhutoryansky]

Thanks for the compliment about my video. Glad you liked it.

[@allenvarughese1163]

Thanks LOVE from India

[@bhawnasuryawanshi2350]

Voice of the narrator 👍👏