Double and Triple Integrals

Remember the good old calculus days, and all that time we spent with integration? Let's go back! Oh calm down, it wasn't that bad. Besides, we won't be learning anything terribly new, we will just look at double integrals and triple integrals, which are pretty much the same as single integrals, we just integrate more than once. But admittedly it can be a little tricky to set up the bounds of integration and choose the order in which we integrate each variable, so let's go through the rules and some examples.
Script by Howard Whittle
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Best teachers so far: 3blue1brown and Professor Dave.

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You seriously wouldnt count sal or Patrickjmt?

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Professor Leonard is the best for me, by far

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Zach star

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For me, the best teachers so far are: Professor Gross, from MIT OCW Calculus Revisited, 3blue1brown, and Professor Dave

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You are such a great teacher. And I like how you use "we" instead of "you". Makes the student feel like they are not alone.

[@MegaYoungscientist]

I think it's great because even the professors still are students at the end of the day, because everyone is still learning more about their subjects.

[@akuo0708]

this is the clearest explaining math video i've ever watched

[@buttersquids]

I was struggling a lot on understanding double integrals on a homework assignment, but this helped massively. Thanks Dave!

[@harishkumar-rn2mw]

Hi Dave, I got confused in the part choosing lower bound and upper bound between 2 cuves. Pls explain..

[@mikepablo]

Brilliant , wish I had this when I did maths so much clearer and simple, during lock down going though my old notes 30 years ago inspired me to study more

[@FlyByPC]

Great explanation! Of course, my inclination is still to hit it with a FOR loop...

[@theresaread6174]

Blessings upon you, Dave. Two minutes in and you’ve already made this make more sense than my instructor.

[@georgefan2977]

Thank you, that was one of the most straightfoward explaination!

[@mastershooter64]

So you can actually find the volume inside a tesseract niceees

[@kek3324]

its a hypervolume

[@JorgetePanete]

@@kek3324 It's*

[@doxo9597]

Jorge C. M. I can tell you have a lot of friends.

[@OneDerscoreOneder]

It’s just all the side lengths multiplied

[@dinamosflams]

You can find both, the volume and the hypervolume of a tesseract. Although I think it's more difficult to grasp the concept of the volume of a tesseract

[@lemonhscott7667]

Professor Dave, thank you so much for these resources! They've been a huge help to me in ODELA and Vector Calc. I'd be lost without you!

[@ActionJaxonH]

This video is amazing. I’m self taught, going through Professor Leonard’s Calc 3 YT series course, and double and triple integrals are coming up in the very near future so I wanted to get a head start. This is totally logical and it just makes sense

[@extremeforlife8563]

Thanks, I checked out the series after seeing your comment. It is more suited to what I am looking for. Really appreciate it man.

[@lukafarkas420]

good job Dave, you have explained this to me in a most understandable way possible, with clear color indications, thank you sir.

[@ejticomon9603]

Why is the y integral from x^2 to 2x and not 2x to x^2?

[@lolzdatguy4987]

Lower relative to y

[@krishlapatra424]

Really nice ... So clear concept.... Thank you professor Dave

[@mkpr757]

A video masterpiece. Grateful for the illustrations, painted a clearer picture in my mind...

[@jocelyneavila6870]

Thank you for the subtitles in English! It makes it much easier to follow-up.

[@tanishq1026]

Thank you so much! I really appreciate this as we don't get taught a lot of concepts and underlying principles in math, so this just makes the subject so much more interesting. :)

[@huijiang3051]

Thanks Dave, you are a wonderful professor!👍

[@melontusk7358]

I've finally done it. Thanks for your help, my good sir. I'm a lot more confident now.

[@aceflamez00]

Wow that was easier then expected. Thank you for explaining it well.

[@sauravkr.mahato]

Great Visualization of the figure formed by double integration Loved it !!

[@cd8836]

Thank you for the videos. They help a lot! I’ve had a mental block about these topics for decades.

[@shaiyongcheng9721]

This video is better than 1 hour of lecture. Tq very much

[@millieh3179]

Can you do a video on Green's, Stokes' and Gauss' Theorems?

[@Universities_Reels]

Could you send me notes of divergence theorem ?

[@ProfessorDaveExplains]

Millie all three of those are filmed, animated, uploaded, and to be released very soon!

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@lulu , your wicked tactics to paint real Christians ain't gonna work, people like you are the reason muslims conflate legitimate love lead criticism from people like David Wood and Robert Spencer with hatred from people like you and richard spencer.

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20UPH512_Geo Raphael Michael What was lulu’s comment about?

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​@@GeoRaphaelMichael *_Bruh, what_*

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Thats the most outstanding explanation I ever got. Thank you so much!

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Nice work... You deserve a big big big thanks... Go ahead Dave brother

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This is probably the best that this video could possibly have been made.

[@andreacondello]

that is amazing how clear you was

[@coop_the_troop]

Man I’m trying to learn first calculus, I was recommended this from a friend in the math field to help, it seems a bit to daunting, I wish I was I just good at math and actually understood what he is talking about most of the time

[@ProfessorDaveExplains]

Start the calculus playlist from the very beginning and all will make sense!

[@chroma9848]

You're an excellent teacher, you truly are. I survived 1st year of engineering with your help. It would be nice if you can make videos on differential equations 😅

[@sine4859]

wow i didnt expect integrals to have this much depth to them... how can you not love math, omg great video btw :)

[@johnsnow3444]

Was so thrilled when i did the double integral the other way from y/2 to (y)^1/2 and got the same answer! strangely that way took up less paper. Thanks for the solid content!

[@ndereratjijahura2048]

Thanks so much for a clear presentation.

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A learner from Tanzania,🤝you are a good teacher ❤

[@thabolikotsi5765]

awesome vid thanks im a fresher ive jusst started self learning ths topic ths will help wth understanding

[@itsme.01]

In the first comprehension, when we move from 2 to 3 in x,then y=x+1 is lower to 2x . Why we consider here limit in y from 2x to (x+1)?

[@Zephyr-tg9hu]

Thank you so much, Dave! You are and always have been a lifesaver.

[@NoctuaStrigiformes]

Integrals are tight. You made it look super easy, barely an inconvenience.

[@someone________2502]

Great video. Even though I only did high school calculus I could still follow what he is saying.

[@mohamedamr6228]

You are a GENIUS...Such a relief to finally understand the bounds depending on variables... thanks for your hardwork really appreciate it

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Thank you sm genuinely

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Professor Dave is the best...thank you very much!!!!

[@annea1537]

Danke sehr!!! 🎊 beste Erklärung! Mit Abbildungen, klar und deutlich. ❤❤❤❤❤

[@darkwing2924]

Thanks for making calculus fun and interesting

[@christianb4679]

Thanks so much! I never understood this in my Engineering lectures, now it makes sense.

[@os9458]

Hi Rosal.can I know is this thing involve I'm grade 12 or higher grade .Because I'm going to do engineering too. Currently grade 12

[@christianb4679]

OsJ I’m not sure, I studied this in university.

[@nosirrahonline1225]

Oh my god, thank you so much for this video. The videos that I get for my class gave me an example where both y and x's bounds were 1 so I didn't understand how she was doing it.

[@mavlonkarlsefni]

I have a test tomorrow, and your video really helped me to understand integration more deeply. Thanks from Russia!

[@user-fo8cu6ef5y]

You're the E of the mc²! Great lecture. Congrats.

[@bryanwharton6692]

“Performing multiple integrations is not all that daunting, on the SURFACE!!” Haha

[@encinaallaineharleya.8586]

We have a calc quiz tomorrow and this video sure helps, Thanks!

[@amaizasafdar9401]

no word to explain your efforts may God reward this

[@TTeach22]

Great thought and too clear fellow style, I apricate your way of explains

[@alexanderkaiser89]

Hi Professor Dave. I have a request that you may find interesting or not, it’s a mathematical one. I have always found it hard to understand “Inter-universal Teichmüller theory”, “The Collatz Conjecture”, and “The Birch and Swinnerton-Dyer Conjecture”. If you would make a video on one of these topics that would really help 😊 I’m studying a Master’s in mathematics in Denmark. I’m sorry for my English, I’m not mothertongue in English. Btw, I love your content.

[@Jack-lp4jd]

Best explanation around. Though im still struggling to understand how taking the integral of the x-y plane in two different ways gives volume. So the area under the curve given from the first integral just happens to also equal the volume?

[@fightinggod03]

whenever my dad checks up on me if im doing work, i just pull up your videos and he doesnt question it! thank you :D

[@thefatcat1997]

I wish i had u as my teacher during my btech...

[@ItIsMe167]

Good Job, Thanks for Sharing

[@kaymoneyalottoww4828]

cool teacher, straight to the point

[@virajpatel2121]

Sir you are awesome ,I love your explanation method. 😇😇😇

[@adristimaulidafauzi4418]

Finally I understand the sense behind these

[@mlop2484]

thank you sir

[@jacobridley8156]

so helpful cheers Dave

[@christosgolsouzidis1286]

I thought that double integrals give the area of the surface of a function and triple integrals the volume of a function respectively

[@tzonic8655]

Και εγώ ρε φίλε,έτσι δεν λέει και ο γκαρουτσος? 😂

[@didyk3574]

Hey dave can you graph this function in 3D f(x,y)=xy/(x^2-y^2) around the point(0,0)

[@mayashomarieh7941]

Thank you so much

[@prateek3828]

I thank you dear sir for mentoring me . 🙏

[@kamalnath4606]

love you professor dave

[@kerryloh5285]

Nice explaination

[@mohitchaurasia7181]

💥👌Nice explanation 💥👌☺️🤗

[@pankajpokhrel2463]

Very well explained..

[@kaluvanhariharan4256]

Highly informative lecture. Thanks Professor . In 7.31 , how did you the get function f(x,y)? will you please explain?

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I still don't understand why someone can dislike a video like this :( You are incredible!

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[@HorizonSpeed26]

Thanks very useful and understandable

[@JorgetePanete]

1:37 I think there are counterexamples to that statement, but it works in general.

[@paulespettia4949]

Could you tell pls

[@eskayML]

I have a question please For the upper and lower bounds isn't the x² greater than 2x and why didn't it get placed at the upper bound instead?.

[@jainyshah3000]

Brilliant! Thanks :)

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Dave the man!

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He does know all about science stuff......... professor Dave Explains ❤️❤️❤️🔥🔥🔥. I Love this😎

[@danielvolinski8319]

14:30 The first example on the Checking comprehension is wrong: the internal integral boundaries should be from x+1 to 2x

[@saurabhsingh-ow7ue]

thank you sir.....

[@official_darick]

Hello Professor, please can you cross check you final solution to the calculation on volume integral: 3x2 - 3xy2.

[@rambhaustark6622]

but volume needs 3 dimensions right? so shouldn't there be a z coordinate as well?

[@STKeTcH]

You the man, man!

[@pakshah5449]

Thanks Great Video

[@falmataaroorroo4478]

oh men i wish i knew you back when i was taking Applied maths 2 in college

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annnnnnnnnnnnnnd you have a new sub/ bingewatcher

[@shahnaznamirafairuza]

hi professor dave, in 4:00 you said 10 is the volume under that given example surface, why does it considered "under" not "above"?

[@smartdoctorphysicist3095]

Hi thank you very much.

[@ramuganesan]

at 07:12 , how do you say that y runs from the lower curve to upper curve? sir i cant understand that point

[@larsjansen4245]

Are triple integrals used in space-time diagram calculations?

[@freezinfire]

Thank you very much.

[@deltaexplorer47]

WOW !! THANK YOU VERY VERY MUCH !!

[@rva1945]

Is there any video that shows how integrals are applied in engineering, for example, for building a bridge, a dam, a building, etc?

[@arkadiuszkapuscinski6345]

Integrals are the most usefull part of math. Engineering is the example for that. Love them.

[@marcioamaral7511]

Don't forget differential equations which model pretty much every dynamic phenomena

[@henryy6918]

Wonderful video.

[@lordyabo9910]

Can someone please explain why the curve y=2x is greater than the curve y=x^2? Thanks in advance!

[@musthafamb1757]

Great teacher

[@paulgbago4586]

hello can explain how you take x^2 as the lowest value for a integral and another integral. what makes them the upper value on y and lowest value on x ?can you make it more clear cause i tried to find why but still cant find out . At first i thought it was cause the 2x function was above the x^2 from 0 to 2. thanks you

[@yuel.5013]

You're right, depends on the sign. Solve sqrt(y) >= y/2, it's true on [0; 4].

[@davidschmidt5893]

Would love to see some diff eq stuff. You’re so easy to learn from

[@ProfessorDaveExplains]

coming soon!

[@davidschmidt5893]

@@ProfessorDaveExplains I’m so excited. I’ve been on a math kick