Double and Triple Integrals

[@akuo0708]

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

[@quakeinboots1744]

Ikr

[@schifoso]

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.

[@GiangTranAI]

Best teachers so far: 3blue1brown and Professor Dave.

[@hzkzg1614]

You seriously wouldnt count sal or Patrickjmt?

[@adityarajsrivastava6580]

Blackpenredpen

[@garethpienaar3119]

Professor Leonard is the best for me, by far

[@gouravgope1367]

Zach star

[@jorge_cazares]

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

[@buttersquids]

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

[@gabrielamorais7185]

This was simply amazing. I have never seen a KZbin teacher be that clear and easy to understand. Thank you soooo much!!! :D

[@theresaread6174]

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

[@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.

[@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

[@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!

[@shaiyongcheng9721]

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

[@FlyByPC]

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

[@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

[@3duaun]

Thanks!

[@mavlonkarlsefni]

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

[@wompwomf]

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

[@melontusk7358]

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

[@Kiky_MedPhysicist]

Thank you sir for your dedication and for making this free! 🙏

[@arjunmanoj9687]

Hi Dave. I loved the video, it was a great explanation and helped me understand double integrals a lot better. I had a question regarding the problem that was solved at the 5 min mark. You mentioned that when integrating for dy first, we need to take upper bound to be x^2 and lower bound to be 2x (at 8:21 mark). But when you solve the problem, the upper and lower bounds were swapped i.e. upper was 2x and lower was x^2. Was there a mistake here?

[@tcy101]

I believe it is correct as it is in the video. I think he means lower and higher in terms of y. E.g. x^2 is lower on the y axis than 2x

[@sine4859]

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

[@harishkumar-rn2mw]

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

[@millieh3179]

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

[@Preparing_and_Testing]

Could you send me notes of divergence theorem ?

[@ProfessorDaveExplains]

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

[@GeoRaphaelMichael]

@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.

[@awesomecodeyay8382]

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

[@ivoryas1696]

​@@GeoRaphaelMichael *_Bruh, what_*

[@werkatfischer3183]

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

[@martind1692]

8:26; why is Y (upper bound) = x ^2. Why choose x^2 over 2x. ---------- And X upper bound =2x. Why choose 2x over x^2? I appreciate the answer☀️

[@moosclesapex4643]

because x^2 is of greater value, duh

[@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!

[@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!

[@georgefan2977]

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

[@cd8836]

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

[@CharlieParis-g8o]

In the example starting at 2:21 does this not just find the area of the surface in the x-y plane that is under the cube? I thought you would also have to integrate dz from 0 - z in order to calculate the volume of the shape?

[@aloischipfurutse5079]

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

[@dellpi3911]

kzbin.info/www/bejne/f5LRmIt3eKuKapo

[@ramuganesan]

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

[@leowalentiny1638]

How do you know it goes from y=x^2 to y=2x why is one the lower curve and the other the higher curve?

[@m4rzb4rz-qq3yq]

Hope it's not to late. Look at the x axis, the one closest to it is the lower one

[@leowalentiny1638]

@@m4rzb4rz-qq3yq no, not too late😅, thank you very much!

[@m4rzb4rz-qq3yq]

@leowalentiny1638 no problem :)

[@sakhamurithanmai4442]

My doubt is the curve which has higher y value at particular x value should be considered higher curve. So it should lie close to y axis I think...​@@m4rzb4rz-qq3yq

[@danielvolinski8319]

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

[@lilpotato5870]

So clear and comprehensible. Thanks!

[@leedsoldat4189]

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

[@4tytude]

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

[@JeanfrankSartori-c9p]

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

[@jocelyneavila6870]

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

[@nosirrahonline]

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.

[@dzdzdzdzdzmr8696]

I dont understand what he meant by lower curve and higher curve when setting the bounds at 7:08

[@bryanwharton6692]

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

[@SaleBabft]

7:39 a mistake in limits integration of 2x and x² ? If 2x on the top and x² under of 2x => if x:1. 2(1),(1)² and if x:2 2(2),(2)² and if x:3 its undefined[ 2(3),(3)³] => 6,9

[@adristimaulidafauzi4418]

Finally I understand the sense behind these

[@cboaitey99]

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

[@cboaitey99]

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

[@JorgetePanete]

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

[@paulespettia4949]

Could you tell pls

[@thefatcat1997]

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

[@TemplarX2001]

annnnnnnnnnnnnnd you have a new sub/ bingewatcher

[@mkpr757]

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

[@MikiSiguriči1389]

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

[@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)?

[@NoctuaStrigiformes]

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

[@AG-ow5fy]

my dear, i finally understand that thing!! u save me!! thank you very very much!!

[@andreacondello]

that is amazing how clear you was

[@encinaallaineharleya.8586]

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

[@laromai2245]

At 3:00 (t=180s), how did 4(xy) become 2(x^2y)? It's the shrinking of 4 to 2 I somehow have missed (although watching 3 prior lessons)

[@uwu.-.5873]

Bro how do you integrate. You add power +1 and divide by that power+1 Thats all he did. x^1+1 then divide by 1+1 which is 2

[@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 😅

[@Melisaakcm]

At 7:27 shouldn't the lower bound be 2x and upper x^2 Prof. Dave? Could you explain how you chose these values? Many thanks.

[@aimanfayyaz3159]

If we look at the graph, the curve y=x^2 is closer to the y-axis hence being lower while the curve y=2x is farther from the y-axis hence being the upper bound. Another way of checking that is by putting values in the curve equations. As X runs from 0 to 2, putting 1 in the equations gives lower value on y=x^2 so it is chosen as the lower bound.

[@Melisaakcm]

@@aimanfayyaz3159 thank you

[@sourashian5901]

If we look at the graph, the curve y=x^2 is closer to the y-axis hence being lower while the curve y=2x is farther from the y-axis hence being the upper bound. Another way of checking that is by putting values in the curve equations. As X runs from 0 to 2, putting 1 in the equations gives lower value on y=x^2 so it is chosen as the lower bound. Also, if you start from the x-axis and then continue going up to the y-axis, the first line you would hit will be y=x^2 hence that is your lower boundary and the second line you would hit would be y=2x hence that would be your upper boundary.

[@Melisaakcm]

@@sourashian5901 thanks

[@lordyabo9910]

@@aimanfayyaz3159 what if there's only one curve and a line that is y=0? the line would be the lower bound?

[@amaizasafdar9401]

no word to explain your efforts may God reward this

[@ejticomon2004]

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

[@lolzdatguy4987]

Lower relative to y

[@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

[@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? 😂

[@cyrex-niaz7798]

Don't they?​@@tzonic8655

[@annea1537]

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

[@derekconn9950]

The only thing is the first graph isn’t really a cube. The base is a square that is 1 unit on the x and 2 units in y but the height of the “cube” is the function of x and y. The way to think about it is like this, the integral uses the volume of a cube equation because in Cartesian coordinates that’s what it is. So like a cubes formula for volume = length x width x height, but here the function you are integrating is the height and the dx and dy values are the length and width of the base of the structure, the integrals limits form the base. Since the integral values are all numbers it forms a square but the height is variable based on the function. If it was just a normal cube the function you’d be integrating would be a number. What the integral is actually doing is creating an infinite amount of cubes all with different heights but the same base area dx dy and summing them all over the course of the square area projected out. The best way to picture it is to graph the function and then draw the square on the xy plane but then picture the square projecting upwards until it hits that function that’s the power of the integrals it allows you to take the volume of not a cube using the formula for a cube by just making infinitesimal cubes and adding their volumes together. What the total volume that we are finding is actually a shape that has a square base and the height is like a parabolic shape. The best way to picture it is to graph the function in 3d and picture the shape of the bounds on xy plane which in this case it’s a square and picture it’s height as the image of that square going up until it hits that 3d graph.

[@someone________2502]

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

[@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?

[@ahmedrazim7567]

Actually at 8:34, we get x = ±√y, x = -√y being the other half of the parabola. So we write x = √y .

[@Silwanos]

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

[@aceflamez00]

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

[@JOHNLYMUELBAUTISTA25]

7:31 Where did the equation for f(x,y) come from?

[@Aaron-q9p4h]

Thank you, so much

[@vinzchannel01]

help please 5:17...... where did y=x^2 and y=2x come from????? When our given is f(x,y) = 3x^2-3xy^2

[@sourashian5901]

it's given to us.

[@prateek3828]

I thank you dear sir for mentoring me . 🙏

[@dalibormaksimovic6399]

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

[@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.

[@sauravkr.mahato]

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

[@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?.

[@acemaster1877]

Triple integrals don't calculate 4D hyperspace. They are just a different way of calculating 3D space when the area of the inside isn't increasing or decreasing linearly. You'd need a Quadruple integral to calculate hyperspace. Look it up for clarification. His reasoning sounds correct, but I double-checked it with a few sources, and they state that triple integrals only calculate 3D space.

[@TTeach22]

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

[@azadprasad856]

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

[@kaymoneyalottoww4828]

cool teacher, straight to the point

[@mujtabanazeer8837]

Thank You very much. But, what do you mean by lower curve to higher curve while setting the bound?

[@SMNAviation]

I consider the view count on this an insult. And an even greater one, that 52 people dare dislike it. Dave, your videos are amazing.

[@thabolikotsi5765]

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

[@JOHNLYMUELBAUTISTA25]

14:47 I got 6x^2 when I substitute 2x to the derived equation for y...

[@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].

[@monhbatonon6333]

now this is what math should be about. Solving something real

[@Natearl13]

But can Professor Dave explain why my prof throws trigonometric bounds on the triple integrals like wtf why do you hate me

[@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

[@krishlapatra424]

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

[@darkwing2924]

Thanks for making calculus fun and interesting

[@wheatlysparble7900]

I wish we had you as our professor

[@kamalnath4606]

love you professor dave

[@rva1945]

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

[@ejimadam]

I don't understand which would be the lower curve and which would be the higher curve?

[@jaccckky8978]

I don't know why I need to pay to go in the Uni after this video

[@neuacc7462]

So they give you a paper stating your watched some yt videos

[@pawanmarahatta5749]

blessings upon you, Dave.

[@joseroces6566]

I don't understand when it's lower or upper bound. I'm guessing it's whichever function has a larger area under the curve?

[@md.azmiribneislam6885]

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

[@salehsalum5308]

I need more details in triple integrals, it is hard to imagine in 4D

[@mackb996yt3]

It is impossible to imagine a 4d space literally

[@mohitchaurasia7181]

💥👌Nice explanation 💥👌☺️🤗

[@stonerambo3982]

I was only confused in one step it got cleared only within first 2 minutes...