Double integrals to find mass and center of mass of the lamina (KristaKingMath)

► My Multiple Integrals course: www.kristaking...
Learn how to find mass and center of mass of the lamina using double integrals. Use the double integral formula for mass and the double integral formulas for x bar and y bar to find center of mass.
● ● ● GET EXTRA HELP ● ● ●
If you could use some extra help with your math class, then check out Krista’s website // www.kristakingm...
● ● ● CONNECT WITH KRISTA ● ● ●
Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student-from basic middle school classes to advanced college calculus-figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: www.kristakingm...
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[@Ashley.nowicki30]

Krista you have single handedly taught me my entire calc 3 course. It is only thanks to you that I am passing! Thank you so much!

[@brandonmalley1134]

Finally a decent video that explains everything simply! Thank you

[@brucelee570]

From the deepest part of a college freshman's heart, god bless you :')

[@kristakingmath]

😊

[@christopherreidthemathands7850]

Thanks Krista. These videos, you post, help me prepare for my tutoring sessions.

[@kristakingmath]

Awesome! I'm glad you're able to use them to get prepared. :)

[@MoAntphonyMo]

Is the X bar and Y bar correct? In the book the X bar is yp(x,y) and Y bar is xp(x,y) so im confused now...

[@hayley3286]

soooo helpful!!! at a pace I can actually understand (unlike my prof) & super clear!! thank you so much, makes my life so much easier

[@kristakingmath]

I'm so glad it could help! Thanks for letting me know. :)

[@EDUARDO12348]

Holly moly, it's test day and I've never heard of double integral! ahhh

[@Ekvitarius]

O_o

[@guythat779]

It's an integral that you do another integral to Not a big deal... my math proff certainly thought so since she gave us that treatment, casually mention double integrals like they're nothing

[@ahmadmax4478]

Thank you for all the videos. Detailed explanation=full understanding , and that your way.

[@kristakingmath]

I'm so glad you like them!

[@kristakingmath]

You're welcome! Glad I could help! :)

[@notadeity]

Great explanation, very illuminating. Thanks!

[@kristakingmath]

+Ben Tulloch Thanks! Glad it could help!

[@Siddeo]

thanks krista..i passed my test because of your videos..

[@kristakingmath]

+SiddheshDe Awesome! :D

[@KingDuken]

Thank you! If only professors could explain things the way you do or Khan's Academy, life would be easier haha! :)

[@ThePinoyMamba]

It looks hard but suddenly its simple! You rock Krista!

[@darrennixon8765]

You're the best Krista!!

[@anonymous1361015]

Very clear and professional! Thank you

[@luisblanco9076]

Watched this minutes before my test. It helped a lot

[@kristakingmath]

Oh awesome! I'm so glad it helped!

[@tyler-simoni]

Studying for my Calc 3 exam later. This video has saved my life lol THANK YOU!

[@kristakingmath]

You're welcome, I'm so glad it helped! Good luck on your exam, I hope you do great!!

[@tyler-simoni]

Just finished the exam. Definitely helped!

[@bleuemoone8710]

Thanks, this is tough but doable thanks to your help

[@sairanjanmuni8698]

Find the mass of the plate in the shape of the curve (x a )2/3 + (y b )2/3 = 1, the density being given by ρ = µxy, µ being a constant. Please do this

[@MrDoubleCloth]

Had to search for solution like this for an hour... Thank You

[@kristakingmath]

I'm glad you at least found it! :)

[@ChompChain]

Thank you so much for this. I never comment on education videos but this was bomb

[@kristakingmath]

You're welcome, I'm glad you liked it!

[@hairulazman1156]

can someone explain how she get the result of 3/4 and 3/2

[@TKO177]

Good video, very helpful. I think it might be more helpful if you explained why you integrated in the order that you did (dydx). Which I think is because this is a vertically simple region? Either way, it took some thinking for me to figure out why you chose the order you did, and it would be nice if that was explained better. =p

[@Mindbulletz]

If you were to integrate with respect to x first, you would have to split the integral up into two integrals, one for the region above y = 1, and one for the region below. Notice that when integrating w.r.t. x first, you have two possible equations as your upper limit.

[@sarindranramayes]

you saved my life.. thank you so much!

[@PhilthyPhilTheOriginal]

you are awesome !! :) your videos coincide with my current class !!!!

[@kristakingmath]

+Philthy Phil (TheOriginal) :D

[@sujitsharma5158]

Crystal clear explanation....👍👍👍

[@KuldeepSingh-wo4yl]

Well explained 👍 👍

[@kristakingmath]

Thank you so much, Kuldeep! :)

[@biswalty]

What program do you use to make these highly informative videos?

[@kristakingmath]

+Tyagadipta Biswal I explain here: www.kristakingmath.com/my-videos

[@ScaryBRIT94]

Great video, but what would the new limits be if say the top coordinate was (1,3) instead of (0,3). What would the new bounds be on y?

[@wildweston]

The p is actually 'rho', and it is expressing the density as a function of x and y.

[@360xi]

Awww Krista! You have inspired me to make my own video for a math problem that I had a hard time figuring out. I went to tutoring. But yea....Im not as professional as you. I can edit myself though. I already recorded I just got to edit and upload. Thank you so much Krista!!! Even though I failed all the quizzes and midterms for cal 3 after seeing your videos I'm ready and I'm going to own the final...hopefully.

[@kristakingmath]

Very cool! I hope you can rock the final! :)

[@Rabixter]

I haven't actually studied this in my first year, but it looks interesting. What does the function P(x,y) represent? Why is it specifically P(x,y)=x+y for this specific problem?

[@Neueregel]

Welcome back. Nice new work.

[@edwarddi3833]

Thanks Krista!

[@kristakingmath]

You're welcome!

[@benpaul8964]

what equation do you use to find the moments of inertia of the lamina?

[@kristakingmath]

Ben Paul This might help: kzbin.info/www/bejne/mJfCoIGAZc94i6M

[@kristakingmath]

it's not bad once you get the hang of it! :)

[@AC-ls9tl]

Are we integrating by dy first in this case because the graph is easier defined by an upper and lower bound or in the case of looking for mass, we will always integrate by dy first?

[@aysalasaysalas8040]

anyone else notice she has one green and one brown eye? Super cool! love your videos btw, helped me ace my final! :)

[@kristakingmath]

It's more like a green spot within an otherwise generally brown-ish eye, lol. :) Congrats on your final! So glad the videos could help.

[@bujocomedy7342]

On the x coordinate you have an error you should multiply with y not x .Can you confirm that or it's on my side

[@v5q211]

I love you Krista

[@sparklingnight7513]

Thank you 💖💖💖

[@user-uw7ms4tt2m]

And what about moment of inertia ?

[@lewismuriithi7220]

thank you very much well understood

[@MRAXELGRINDER]

Awesome, thank you, great

[@toosikkid123]

This was great!

[@kristakingmath]

+toosikkid123 Thank you!

[@stvs9722]

what program did you use to write on the blackboard?

[@matematica_do_zero]

Obrigado. Muito bom.............

[@kristakingmath]

Thank you so much!

[@DakshChhokra]

It's Brazilian for "Thank you very much"

[@matematica_do_zero]

Good video...

[@ayukibrahim628]

whether the lamina is referred to in this video there is a section on the bone?

[@brokenflightz]

I'm confused. My calc 3 professor said Mx = integral of the region yρ(x, y) dA and My is equal to integral of the region xρ(x, y) dA and X bar is equal to Mx/m and y bar is equal to My/m Is there a difference or will I still get the same answer if I do it both ways?

[@guillaumethemapler]

brokenflightz the My /m =x My =double integral x*p(x,y)da this should be it it think

[@skoobiesnackk]

x bar is equal to My/m and y bar is equal to Mx/m, and My = xρ(x, y) dA and Mx = yρ(x, y) dA

[@danielgood]

yes do it the way your professor told you. Also do not integrate using dydx, you need to use dxdy

[@fisheatbrocoli3696]

y=Mx/m in class in just the same as here... it is just that she already insert it in the equation in the first place. So her equation become X = 1/m ( MY )

[@ntuthukomthiyane5017]

one of the best, thanks :)

[@kristakingmath]

Aww thanks! :)

[@amonevelina8261]

this was very helpful.

[@kristakingmath]

Great! :D

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Your a Queen, thank you :)

[@edwinsebastianperezrodrigu8659]

OMG she gave you a heart

[@arbitragezurich9463]

@@edwinsebastianperezrodrigu8659 yes I got so happy :) she has been so helpful to thousands of students literally:)) she is so humble and considerate

[@mhdwhd3442]

I really appreciate what you do.Thanks

[@alperenustun7094]

Thank you!

[@rockstone7851]

Thank you

[@kristakingmath]

You're welcome! Glad it could help.

[@maddybritt264]

Your channel is amazing

[@abhay_hegde]

Thank you very much ma'am...

[@kristakingmath]

You're welcome! Glad I could help.

[@manzoorboy06]

Thanks!

[@thantzinmaung-yq6cu]

Cant we just use y boundary from 0 to 3? Why describe it in terms of x?

[@MrFbi101101]

this is just great!

[@kristakingmath]

+MrFbi101101 Thanks!

[@maazmustafa2322]

Thank You so much!!!

[@kristakingmath]

You're welcome! :)

[@khushalkhushal6835]

A lot of thanks mam

[@persianshawn92]

you don't happen to have any videos for finding the centroid/center of mass with the help of triple integrals?

[@kristakingmath]

+Shahin Sateei You mean like this? kzbin.info/www/bejne/lYS8gKyfm5Kcqqc

[@persianshawn92]

exactly, though that example was too easy :/ was hoping for something along the lines of "centre of mass for a solid in the 1st octant" etc.

[@kristakingmath]

+Shahin Sateei I'll try to add something like that. :)

[@Armedon25]

Thank Youuu !!

[@karangujar317]

nice solution

[@pinkfloydeagles34]

do you mean the general form of the double integral?

[@Imasikufteakiejr]

thnks

[@kristakingmath]

you're welcome!! :)

[@myside7]

i want to know how to solve definite integral when the intervals is variable

[@myside7]

thanks

[@bemoentertainment5046]

thanks soooooo much

[@kristakingmath]

+Ahmed Hamada You're welcome!

[@bemoentertainment5046]

Please,Do You Mind If You Explain Moment Of Inrtia .?

[@shermandemetrius1696]

I don't understand the power rule used for y= 1/2y^2. Please help.

[@shermandemetrius1696]

Haha! Sorry! I got confused with differentiation and integration. I'll cleared up now. Thanks a lot for the video. I just wish I'd have seen this sooner.

[@kristakingmath]

thanks!! :D

[@govind22703]

Thank you, your really good in math 😂:)

[@kristakingmath]

:)

[@nhunguyen9393]

Thank you very much. :)

[@kristakingmath]

You're welcome!

[@tronghai55]

You are certainly the best teacher in calculus 1 and 2 but basics me she is a must in engineerings but some must have at least more than 7 years to become a nerd .in pascal maths is boring to death there is no logics maths the intecracies is just a weird mind bending .but creativity some had to have at least 6 to 8 years to pretend to be something unez to consider .even ivy are just ivy in their classical ways respectable.dimetimes is becoming boring in the sense of research and creativity .when you look at rafale or airbus 360 even shuttle it's me who design it and the crook Italian stole by mass murders by crooks lawers of that Rumsfeld ..and panetta the 2 make the pair .why we build rafele and airbus cause some nutties engineers with no purposes are out of jobs .just like I build the Lumineers to the president dean of fr professors cause our tenure masters in dentistry is out of job with no reason to be blundgoen by some nutties apartheid progressiste haters .dime just can't get it get yet that once the crooks are know by their tenuteoutrage if haters and prejudice the good ones are neither to fearcause some nwrds are recin by their peers and not the crooks mobs crial intentions of lavish underseval haters .

[@alperencakici]

thnx

[@AbhinavShresthaTheGreat]

why is p(x,y)=x+y necessary to know to do this calculation?

[@fogijay]

p(x+y) is the density which in this case is mass per unit area (density = mass/area) (and is not necessarily constant over the entire area). so you integrate p(x+y) with respect to area to get mass. (one could see that from mass=density x area)

[@TheO2tenify]

MVP!!!

[@brianboks878]

amazing😍

[@GETURHANDSUP916]

I have been saved.

[@flarosantana6330]

Will the mass always be positive?

[@donotbebiased6987]

Yes

[@tyan4380]

Amazing

[@kristakingmath]

+尾生子煦 Thank you!

[@SuperStruct]

Why tho?

[@SuperStruct]

The only thing stumping me is the comprehension of the integral being a weighted average of the values, understanding x and y bar. The equations aren't intuitive. The only intuitive equation to me was integrating density P to get mass.

[@kitordona7303]

i got = 1/2*x^2 +xy

[@transcendentalistone]

Beautiful and smart, what more can you ask for?

[@mayanksingh2298]

thank you ;)

[@kristakingmath]

You're welcome! Glad I could help.

[@v5q211]

Never knew Americans are good at maths

[@anikoustos2]

ρ should be used for triple integrals σ should be used for double integrals I think.

[@greatsea]

+anikoustos2 ohhh, how anti-establishment of you. edgy.

[@TrueYears]

My queen...........

[@rebwarmohammed7644]

good

[@kristakingmath]

Thanks!

[@rebwarmohammed7644]

What is your name and name facebook to help me ...Iam student master mathematic

[@AngusHollands]

Rebwar Mohammed smooth

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What

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[@a.b.h.j.t.c3136]

I love your voice

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I love you where are you from?

[@mithilamoontaha7127]

big fan

[@kristakingmath]

Thank you so much!

[@hg2.]

Some new concepts here (roe?), but will take on faith.

[@mahmoudshaeer7194]

super like (Y)

[@pinkfloydeagles34]

p(x,y) is the function for the z points of the surface. Just like y=f(x), z=p(x,y). I think the person who made this chose p(x,y)=x + y because of its simplicity.

[@nooreistalking829]

hi, i'm a BA Math major at West Virginia University and have passion to start my own college, let me know if you're interested cuz, I'm interested in educating people rather than robbing them of their own money

[@darekced]

I like that you do EXACTLY the problems we have to do for homework. How do you know? I have a suspicion that you are a witch ( ;