Calculating Double Integrals over General Regions

Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) / patrickjmt !! Calculating Double Integrals over General Regions - Two complete examples are shown. For more free math videos, visit PatrickJMT.com

[@himomhey3437]

this was uploaded almost 11 years ago. Crazy to think that the people who firstly watched this video is now probably a licensed mathematician,engineer,is in the business world, or even dead. I was only in 2nd grade when this was uploaded, now im a freshmen college. Holy fck

[@dibblesvillard5268]

dang I was in first grade or kindergarten

[@Aaron-lp3zt]

This video brings me back

[@reshankjain6622]

I am here in 2021 in my first year of college and Im watching this now XD

[@echo.1209]

When this was uploaded, I was in kindergarten. Now, I'm in my second year of university.

[@franciscoanaya7345]

did u graduate already

[@TheOne-tl9yb]

The video really helps. Btw at 8:45, it should be x^4/2 instead of x^4/4

[@HarpreetSingh-ke2zk]

Silvanus P. Thompson, F.R.S wrotes in "Calculus made easy, 1914, "Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the text books advanced mathematics-and they are moslty clever fools-seldom take the trouble to show you how easy the easy calculations are. On the contrary they seems to desire to impress you with their tremendous cleverness by going about it n most difficult way".

[@patrickjmt]

@nikan4now in this, case it represents the volume below the surface and bounded below by the x-y plane

[@nevmiku]

Thank you so much! This old video right here helped me through how to find the limits to calculate continuous random variable joint probability density function. Thank you again!

[@BombedNevada]

Bless you! Who knows how many engineers, physicists, statisticians, chemists, etc. you have helped. You the real MVP, Patrick. Thank you!!!

[@AjaySharma-dl7uy]

Your are prolly an engineer by now

[@BombedNevada]

@@AjaySharma-dl7uy I build and deploy statistical models for forecasting product demand (supply chain) funny to look back at this comment. Things have gotten a lot better.

[@AjaySharma-dl7uy]

@@BombedNevada oh man ! Do you have an Internship position available lmao

[@LupoBHS]

What I hate about teachers is they teach and quote everything out of the book. They don't take the time to make sure students understand what the book is saying. That's why I rarely go to class because if they are just going to spend class time citing the textbook, I can do that on my own time. That's why I love Patrick's videos, they give real insight to what textbooks are trying to communicate.

[@inuyashalivesbaby]

OMG.... I love you!!! This stuff scared me and confused me so much, but you make it easy peazy in 10 minutes or less.

[@user-rb4bv8st5q]

Thank you so much. I understand my 3 hrs. lecture in about 10 minutes ! 🙏🙏🙏🙏

[@patrickjmt]

@Tkdkid9 ok! many of the comments point out the mistake i do believe

[@patrickjmt]

ops, yes you are correct! i just added annotations so that everyone else will know! thanks for helping!

[@o0Evangelion20140o]

Wow that was awesome, I am taking calc III this semester and I did the first integral myself after a few seconds and got the correct answer! Basically I jumped ahead several sections and did this; great job!

[@xijieliu4996]

Thank you for a lot of videos that help me get a cracked-open point to understand some concepts.

[@MrAlbashiri]

As always, Thank you for the amazing video.

[@mkrasmus]

Excellent. I just realised that you when you get integrals in the form int x dy you can treat the x as a constant and throw it outside the integral before integrating whatever is left inside. Big aha moment - thanks very much!

[@trudal987]

Hot damn patty, youve done it again! Teach me in 5 minutes what my prof couldnt teach in 5 hours. Like a boss.

[@muhammadnasihi2496]

Thanks Patrick. U makes my college math easier. Im going to have my final next week. Btw, i already subcribed and liked the video. Looking forward for your other videos. ;)

[@patrickjmt]

@martmelee thanks ; )

[@kubrayurtgezen7368]

you are an amazing teacher ! I 'm studying my exams always with ur videos , short but meaningful

[@zgr9]

you are very good at explaining this, i wish my calculus teacher was this good

[@warwick802]

Dope how math and physics haven't changed since 2008. These videos should be able to help someone from the year 2123

[@RbtV92]

I mean, the math is amazing but what's mind blowing is the fact you do all this stuff in sharpie and make like ZERO mistakes!!! great vid, Patrick.

[@sarotwileri2114]

You are amazing I don’t know what I could do without u 🌺

[@patrickjmt]

i just noticed that too! i need to learn how to do some wide screen filming now

[@patrickjmt]

@MrKyte12 ha, you can try : )

[@krishan]

you sir, are a freakin life saviour!

[@zgr9]

mmm memories, im currently taking an electromagnetics course and we use a lot of double and triple integration, surface/line/volume integrals. fun stuff!

[@Kurtlane]

@NicoThR, 3/10 is correct. The error that lead to 7/20 occured at 8:45, where there is a comment 'i write 4 in the denominator, but it should still be a 2.' It should be a 2, and with a 2 in the denominator you get 3/10.

[@AngusTaoHuang]

Thanks for interpreting the specific procedures of processing fist and second integral. :)

[@MinhieMinh]

You are seriously the greatest person on the face of this planet.

[@xfootballer81]

You probally think i'm stalking you due to the number of videos that i say this on but here it goes again...THANK YOU! Also you are amazing and are the reason why i like math and am thinking of adding a minor in it. It's fun when its done right and there's a good teacher, like you, to teach it.

[@victorchendra4282]

It really help me doing my homework, thanks a lot

[@martmelee]

The best!! nothing else to say man, you are the best.

[@ayberkgorgun]

That helped so much, thank you. you wrote (x^4)/4 at 8:44 but shouldn't it be (x^4)/2 ??

[@zaidalzeiq9840]

Ya ur ryt he has mistaken somehow

[@SNYPQuiCKsHOTS]

sheesh though i was crazy

[@aymenelmoubarik6332]

@@SNYPQuiCKsHOTS you still are tho.....

[@patrickjmt]

@TokeItUpTyler well, tell that to the person in one of the other videos that was bitching cause i derived something. his comment: this is a useless waste of time, we can just memorize the formula. who cares about deriving? and i think most teachers do try to explain things the best they can. often times students do not ask questions so the teacher is left to guess whether or not they are getting through.

[@hitmanLis]

Nice!!! That's exactly what every math student need

[@Backlash2224]

Thank god there were comments about the mistake. I was following along on the website, and when he got x^4/4 I was trippin for like 20 min trying to figure out what was going on lol. Not baggin on ya man. I thought I had done something wrong considering ur videos teach me most of what I know about this stuff.

[@loopingmoon1444]

Great video!

[@user-jr9tf1iq8e]

Thank you very much, Sir🙏🏼

[@daoneproductions]

The answer to the second problem is 3/10. Patrick made a mistake at 8:44 by changing the 2 to a 4. It should be x^4/2.

[@muddwell]

Thanks dude, amazing how much study time you can save by watching a video.

[@CathyMahloana]

Man thank you so much!! I fail the topics you dont cover.

[@mk_30th]

You make calculus transparent!! Thank You!!

[@Tkdkid9]

@patrickJMT Ok, no problem at all =) I was just curious. Thanks for your comment and for everything you do. I really appreciate all of your videos!!!! I would be completely lost with out your help!! You are so awesome!!!!

[@aboodation]

You are a live saver!

[@AriKariG]

you are amazing pls keep making vids

[@Avirajjaulim]

@patrickJMT Thanks Mate !I guess these people are just not willing to admit taht this was actually one of the best lectures about double integrals over general regions (:

[@mpho212]

thanks for the video it really helped me

[@alimustafasherwani9131]

thank u man really appreciated ur concern ur damn easy 2 understand

[@8749236]

final ans` should be 3/10 i believe... (since he mis-wrote 2 as 4..)

[@buriza2401]

Thanks a lot sir.

[@kjskkr]

thanks ,very helpful

[@sheruloves9190]

Thank you sir

[@nirajjavkar]

You teach better than my college teacher THANKX !!!!!!

[@jacobroberts8931]

my multivariable calculus teacher sounds like dr doofenshmirtz an jarjarbinks had a baby. i cant understand a word he says. thank you for this

[@DATO756]

it is great thanks for the videos

[@TheMoohoos]

thank you for everything

[@yashwantsingh5328]

simply u made it easy.. thanx

[@patrickjmt]

thanks : )

[@nusaifsajith6397]

THANK YOU SO MUCH SIR!!!! LIFE SAVER

[@lilwayneisdabest1]

this was so helpful

[@hackersvirus90]

You saved my life man :D THX

[@SciOmOs]

Patrick ... I really wish you were my Math teacher .. I think my life would have been so much easier ! =D ... Thanks for your great vids :)

[@muhammed.bariss]

Quite helpful, thx

[@patrickjmt]

click on the upper right 'x' to close them!

[@HongZhiLoo]

Nice. Thanks

[@MrKyte12]

can I subscribe twice? :) your video are so amazing.. there is nothing I can say but thank you so much.. Thank you for helping us..

[@nicomarchantlara]

buen aporte. its a great video, thank you very much.

[@surpriseqwenta3473]

Could you please drop a video on those jacobian what what.. i really like how u simplify things

[@bezae]

great! thank u mate!

[@patrickjmt]

@TokeItUpTyler i wonder if that is what my students use to say about me... my guess is: yes.

[@emiliablack5269]

thanks a lot bro

[@AshanPeirisNCVU]

Thanks man

[@DGEKuShOM]

you would evaluate just as if it were a constant. if it happens to have a variable left in the answer after your final evaluation I would suggest switching the order of integration or try using Fubini's Theorem

[@user-pq1cj3hy3q]

thank you!

[@codiehennigan6007]

Only because I didn't see it in the top 10 comments but on the last equation he doubled the denominator when he did the last part of the first integration which makes the last problem 3/10 not 7/20 but other than that this video was a big help thank you, cram night lol

[@moizkhalid2714]

really helpfull video..just made ny day (:

[@Vidk1ng]

Wow this taught me more than my math lecturer did in 2 hours.

[@motivationiskey5694]

the answer is actually 3/10 after the mistake was corrected

[@moizkhalid2714]

+Jaydev Thompson yes

[@motivationiskey5694]

yea np

[@mohammedmarzooq96]

Thanks for writing that down. I was already going to smash my screen after seeing the answer that Patrick wrote, lol.

[@motivationiskey5694]

Lol OK ...

[@mickeymoose9368]

I started to second guess my answer until I saw the comment, thank you :)

[@ruchieru]

When you're substituting U into the integral, you're basically integrating U as a variable. If the original limits of integration were let's say 0 to 1 for X, then when you're substituting X as some function of U ( u=x+1 or something), then obviously your limits have to change as well, since your variable is not just X anymore.

[@miggy53092]

gee, ur great!... keep it up!...

[@drsrsr]

thanks alot

[@TudyZZZ]

Thank you so much all teachers should learn from you rather than being proud and arrogant

[@bukhariyusoff3082]

thanks!! help a lot!! :-D

[@imankaid3228]

in the first example when u replaced by u where do you take the numerator x??

[@dcoophx]

I did this exact problem today in class, the final answer is actually 3/10 (when you adjust for the mistake in the video, which PatrickJMT has pointed out for us in the video).

[@squishfishtigerhen]

How do we know if it's easier to integrate with X first or y first?

[@nikan4now]

Great video. Does the double integral of F over a region have a physical interpretation? Thanks.

[@RitavonLex]

I had a conversation about you in school today. Everyone seems to love you. lol

[@inycxchris]

@KingOfShadows92, in the outer most integrals, the limits of integration MUST be constants. The limits of integration in the inner integral can contain only 1 variable . And when doing triple integrals, the inner-most integral can have limits of integration that contain at most two variables. Correct me if I am wrong I need to be sure for my test.

[@Paskovitchz]

Hi Patrick, I was asked to do a similar problem in a recent exam. I was asked to do the double integral of x over some limits and was shown a rectangular shape on the graph, from which I had to set the limits myself. My question is, does it not matter what expression you are asked to integrate?

[@starter3200]

@odoc01 He didn't have to plug it back into the original variable because by changing the limits he changed the integral. If he had kept the same limits then he would have had to switch the variable back. Good luck on your midterm!

[@odoc01]

@starter3200 Thanks so much! Makes sense now :) Cheers!

[@josephozdemir6210]

Thank you! May God help you in this and the other world

[@pcs06d]

Final problem should be 3/10.

[@Biscuit9891]

Finally, I knew it. Exactly, because he should of gotten (x^2)2/2 = x^4/2 and not x^4/4. Then when integrated, you should get x^5/5•2 = x^5/10. At the end it should be 4/10-1/10 = 3/10 and NOT 7/20. Good thing I look at the YT comments after problems, just to clarify his mistakes.

[@Biscuit9891]

I meant to say (x^2)^2/2. ***

[@pcs06d]

No worries, dude; always double check work. Both insanely smart professors and elite textbooks make mistakes sometimes so always ask questions and check what people are saying.

[@motivationiskey5694]

+Cam Staubus yes 3/10 is correct

[@carlosz2346]

Right before I click on one of Patrick's videos I can already hear him saying, "Alright, iiiiiiin this video..." lmao

[@HDitzzDH]

The answer to the second example should've been 3/10. The mistake happened around 8:42 when you wrote "x^4 / 4" when it should have been x^4 / 2.

[@YaraMel]

take x as constant. if u were integrating 3 u would get 3y. x turns to xy. integrating y^2/2 basic rule.

[@sevag00]

No, since the integral is a multiplication you don't add a y unlike addition or subtraction. For example the integral of xy wrt to dx is x^2y and not x^2y^2.

[@miggy53092]

gee, ur better than my calculus instructor! x10....