Calculus 3 Lecture 14.6: How to Solve TRIPLE INTEGRALS (Along with Center of Mass and Volume)

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Calculus 3 Lecture 14.6: How to Solve TRIPLE INTEGRALS (Along with Center of Mass and Volume): What Triple Integrals mean and methods of solving them. Also discussed is the idea of a triple integral representing mass and volume of a 3-D region. Center of Mass is also covered, but more examples in 14.7)

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29 сен 2024

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@junior5146 3 года назад

30:00 Z-simple 40:30 Ex 1 47:00 Ex 2 58:00 Ex 3 1:19:16 Ex 4 Finding mass with p(x,y,z) 1:36:00 Ex 5 Using Y-simple. Example when you are not sure which order the upper/lower bounds are 2:02:55 Ex 6 2:36:22 Ex 7 3:02:00 Ex 8

@invincible2914 2 года назад

3:15:00 center of mass 3:22:52 Example 9

@receperdogan9285 2 года назад

thx for helping

@deemgr7 Год назад

Thanks

@realslimshady1100 3 года назад

That moment when you realized that you'd rather watch a free video thats 3x longer than your college class to which you've paid a shit ton of money for.

@mahmoudm451 6 лет назад

I got a 97 in Calculus 3 thanks to you professor!

@guitarttimman 4 года назад

pretty good buddy.

@stoppls1709 2 года назад

damn bro

@kalebscott282 2 года назад

I'm here because I'm at a 67% lmao

@canman5060 2 года назад

Genius !

@johnathantran1101 2 года назад

97 seriously! you are a god's child

@JML1987 4 года назад

"Yo dawg, we heard you like integrals. So we put integrals inside integrals inside integrals so you can integrate the integration while you integrate." - Leibniz

@theshoulderofgiants 3 года назад

Newton**

@anthroporraistes_ 3 года назад

@@theshoulderofgiants Leibniz***

@nebaashibruce8793 3 года назад

dis sound like a yaad man mek dis comment.

@canman5060 2 года назад

I integrate 1500 times in my Phd Maths Thesis !

@meshalalshehri9207 2 года назад

@@anthroporraistes_ yuiireuewtwtowwuty I’ll errands e We yeuu

@siddheshss12345 6 лет назад

I was looking at my phone when he said "look at the board, stop what you're doing and look at the board" so I put my phone down and did what he said.

@etorr19165 8 лет назад

All of your videos are super helpful. Great way of teaching and explaining the material. Thank you for taking the time to also teach us, by posting these videos .

@giei4515 8 лет назад

I've always wanted to say that I do not know how you do it but your English is so clear. I come from Polish and my English is at the level of understanding and you listen to great! You are an excellent teacher. Even the language barrier does not spoil the content. Thank you for your videos and knowledge. Yours faithfully

@chandranichaki9580 2 года назад

Watch the latest video of wishing happy new year in the language of Mathematics ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-lqVEgVea4Dg.html 😊😊

@John-iy1xi 3 года назад

You know what I love about Professor Leonard's teaching, he squeezes a reminder of past vital concepts and ideas from previous topics. So if you learned, but didn't fully understand something entirely in i.e. double integration, you wouldn't need to go back to his double integration videos; he catches you up on the go. Like the level of in-depth teaching is far beyond what any of my past/current lecturers do, so coming from any other lecturer to videos literally fills those knowledge gaps.

@boraerdemkaratay7243 Год назад

Dear Professor Leonard, Thank you for your explanations. greetings from Slovenia and Turkey!

@hwp842 11 месяцев назад

professor leonard, you are an amazing teacher. my current calc 3 class sucks, and i understand everything so clearly thanks to you. thank you so so much!!

@tommychong3269 2 года назад

I’ve been scrolling RU-vid for so long I ended up here. May god bless my soul.

@chrispinbgwoni8642 6 лет назад

This man is really amazing........thanks Professor and please continue all your hardwork helping people..........

@shadanjaved8827 5 лет назад

you are really good!!! way more good than my professor!! loved your teaching style... lots of love & keep it up!!

@deanharris7149 Год назад

Man! This guy is great at what he does. Makes me wish I could go back to school.

@nigistgebremeskelll5314 6 лет назад

i love u professor leonardo toooooooooooooo muchhhhhhhhhhhhh.thanks alot am a first in calculus in our class .from ethiopia

@reganmian Год назад

Professor Leonard:: "We spent 6 hours doing double integrals"... my class: "we spent 75 minutes doing double integrals"

@SamuelLee-gw6wr 2 года назад

There are two versions of calc 3 in my uni and I am taking the easier version suited for engineering and most business students. But these vids are for the harder version, so I am going above and beyond on such vids. I only need an average of 1.52 attempts to solve a calc 3 webwork problem correctly, but for calc 2 without Leonard it was above 3.00. And now I have calc 3 as the course to pull my grades up. So many of my friends are struggling but I am breezing through.

@sergioandres8041 8 лет назад

Happy birthday Leo :D Thanks a lot lot for this videos

@jesse_sukh1785 7 лет назад

You are Mr. master mind sir......Luv the way u explaining concept.

@puffer_fish58 4 года назад

teacher, you are amazing! greetings from rio de janeiro, brasil, federal university of rio de janeiro

@hasithrathnayake7569 5 лет назад

Helped me understand Electrostatics! Thank You Prof Leonard!

@abdelmajid3409 4 месяца назад

Thanks a lot, you are a very good teacher

@mariambobat 6 лет назад

Thank you. Thank you. Thank you. I am yet to see your teaching style matched. Much appreciated from Zambia.

@not_Enlee 2 года назад

This man can beat you up when you're not listening to his class.

@isabellaz1035 6 лет назад

why can't my math prof be like you... Thank you Prof Leonard!! (my prof spent 2 weeks on this and still made it confusing)

@alyssabrantley5073 5 лет назад

"I trust that you guys can learn this stuff really well. I know that you can" 39:10 Tucking these words of affirmation inside my pocket...

@solinsaadi2473 6 лет назад

thank yoy so so so much ❤ l'm watching your videos from the first second till the final second l can't tell you how much you helped me l really appreciated ❤ much love for you professor ❤

@harveyfloyd6268 4 года назад

there's a mistake at1:59:50. the integral should be 8r-r^5/10

@TUMENG-TSUNGF 4 года назад

he was distributing the r into 8-(r^4)/2, he hadn't integrated it yet

@TheFinalStanza94 4 года назад

Professor leonard dropping an f-bomb at 1:53:18?? hahaha idk why but that made my day... we're being taught by... one of us :')

@alexrogers9889 4 года назад

In the example @ the 2-hour mark, Your final answer comes out to 64pi/3. That is incorrect. You converted the 16 incorrectly in the final integral. Only scaled by 2 instead of 3. The correct final answer is 32pi.

@yaqinalbirawi5634 7 лет назад

For the exercise at 1:21:00 , are we restricting the region to be in the first octant because we are looking for the mass? Like if the question was just to find the integral would we still just integrate over the first octant? I don't understand why we limit it to the first.

@yaqinalbirawi5634 7 лет назад

is it because p(x,y,z) = 2z and that has to be positive because its the mass density function and then since 2z has to be positive then z is positive and then for z=2x to be positive, x has to be positive and y=x which means y is positive as well?

@SomeBarrels Год назад

Bro I went to sleep with autoplay on watching a Dark Souls Lore video, how did I get here?

@elizabethmercado4528 5 лет назад

Hi! What happened with 14.5? (I'm from Mexico City and all your videos are so useful!!! Thanks for sharing

@zachshields9913 5 лет назад

That’s just surface area, you don’t need to cover it if you’re doing 15.6 I believe

@deepaksree6 7 лет назад

Time 1:45:13 ,if we plug in x,z=0 ,we get x^2+z^2 as the lowest region and y=4 as the highest . But as you said if we are free to choose any point within the circle ,then any point greater than (1,1) for example pick x=1.2,z=1.2 or x=2 z=2 would give x^2+z^2 as the highest point and y=4 as the lowest. So how to solve this problem?

@kayleew821 5 лет назад

you are the best

@willygilly5292 6 лет назад

"Ever seen a mass without any thickness"? uuh, I mean a paper is the closest thing to i. Love your video, hope you do more in the future.

@منوعات-ت6ف 2 года назад

Single integral can also give arc length, why not mentioned?

@k9thedivine397 3 года назад

love you bro

@fjordfish3363 3 года назад

I leave youtube running while I sleep, and I wake up to this. what the f...:????

@sbk1398 8 лет назад

if the function was P(x,y,z)=x, then would u still do the same thing but changed x to r cos(∅) and continue with the integration?

@babymoet2474 2 года назад

The second time I’ve woken up to this lol

@ISLAMguidanceful 8 лет назад

Happy belated birthday!!

@uxoy 5 месяцев назад

30:00 Z-simple

@engomar98 4 года назад

Please, can you attach exersises. I mean I want work sheets.

@Niko-fj4mo 4 года назад

2:24:25 why can't it be z < 0 and therefore - pi/2 < theta < pi/2 instead of 0 < theta < pi/2?

@samiambl917 2 года назад

I have the same question

@johnsmith-jx8qh 7 лет назад

Hey P.Len Any chance you can do a diff equations or complex analysis playlist while you are cycling the juice? I wouldn't mind a bit of roid rage mid lecture to spice things up a bit.

@teymurasgarli9506 3 года назад

where is 14.5?

@gavingalloza6962 2 года назад

i just woke up to this

@victorhernandez-eg7wp 4 года назад

Dan why is Avery Bradley coming off the bench

@tammyxu3044 7 лет назад

I wonder where he teaches. Those kids are so lucky.

@elida6366 6 лет назад

World record beated - 3 1/2 hours of class 🏅🏅🏆🏆🤔🤔😥😥😑😑💡💡🥊🥉🥈

@ronaldborrell7338 2 года назад

This triple integral video was almost 3:33:33

@Layan-jx4hz 5 лет назад

wow thank you!

@djswagmac7763 7 лет назад

Do you teach linear algebra?

@purewaterruler 8 лет назад

Damn it, why couldn't the video be 2 seconds longer.

@Nicoder6884 8 лет назад

You mean one second?

@gwaites6329 8 лет назад

hha yea either 2 seconds longer or 1 sec shorter, or even 5 seconds longer so we get 3+33=36

@gwaites6329 8 лет назад

X D go to about 1:15:42 and put at .5x speed X D sounds like a high californian surfer

@gwaites6329 8 лет назад

1:15:39 *

@tlhomotsemoteme2423 5 лет назад

G Waites lmfao

@adamt6024 3 года назад

I owe my A’s in calc 1, calc 2, and now calc 3 to professor Leonard. Professor you are a fantastic teacher and have set the bar for all professors across the globe. Thank you for being an amazing human being and being their for me in my upper division math. I honestly don’t know where I would be with out your videos.

@hassanaftab6219 3 года назад

Which book Professor is using!

@irenewilliams5145 2 года назад

That's cool

@evanjones9602 Год назад

@@hassanaftab6219 Early Trancedentals, James Stewart

@ILikeWeatherGuy 7 лет назад

my university goes 3 times as fast as professor leonard but 20% as deep. thanks leonard! :)

@Anigmus_ 2 года назад

@@stillfranc that was pretty good

@johnestrella4223 2 года назад

@@stillfranc o

@bawol-official 2 года назад

Same here, its par the course though for most places that arent prestigous or private schools. Thats not to say you cant find them. The best math teacher ive had in my life was taking a College Algebra course at my community college, flunked math all throughout highschool and passed with an A. Changed my major to Computer Science just so i could learn more math while staying true to my experience in tech. Theres Prof. Leonards out there its just a matter of chance.

@mplsboi1990 2 года назад

@@stillfranc you win the internet

@naoremningthemton5316 2 года назад

@@Anigmus_ ,

@Je66384 4 года назад

This guy saved my semester. My calc 3 professor goes so fast during class that I don't understand the notes I take.

@keylanoslokj1806 2 года назад

Yeah cause colleges try to give you overly compact knowledge in just 4 months

@mewos838 2 года назад

@@keylanoslokj1806 my college: in 3 months :_\

@adamt6024 3 года назад

Now I just need a physics professor that teaches exactly like you and by that I mean easy to understand and wonderful at teaching.

@miguel1502009 3 года назад

Walter Lewin is your guy

@joeyGalileoHotto 3 года назад

Michel van Biezen is a good option

@sergeirachmaninoff6397 3 года назад

Walter Lewin is as good or even better with his demonstrations! Regarding explanations they are on the same level.

@Lena-zn4re 2 года назад

Looks like I saw this comment just in time, physics is absolutely screwing me over

@ritesha8050 2 года назад

sir Walter Lewin is a physics god. There is no way you cant understand.

@cfraijo4898 8 лет назад

THE LEONARD!! Simply awesome. Math isn't so bad when you've got the Leonard in your corner. It's actually kinda fun! Thanks Professor. Not too many people can do what you do.

@masoomaibrahim3893 2 года назад

8th

@upriiserii5895 2 года назад

@@masoomaibrahim3893 nice IK Iiikimonk kk I hi m M I’m MI’mi just Jii hiiinning Ok I’mHhiihi iiinji I’m Ji Hookiiikiio Iiiij I H Iiiiiii I’m I’m

@RakeshYadav-tj5os 2 года назад

@@masoomaibrahim3893 ijjjj J Jjj Jji Jjjjj J J I Jj Jjjjj Jj Jjjjjjj Jjjjjjjjjjjjj Jjjjjjjjjjjjj Jjjjjjjjjjjj Jjjjjjj J Ijjj Jjjjjjjj Jjjj J Jjjjjjjjjjj Jj Jj Jjjj J J J Jjjjjjjj J Jjj Jjjj Jjjjjjj J I j Jjj J Ji Jjjjjjjjjjj jjj Jj Jjjjjjj Jjj Jjjjjjjjj Jjjjjj Jji J Jj J Jjj J J Jjjj Jj Jj Jjjj Jjjjjjjj Jjjjjj Jjjjjjjjjjj J J J J Jj Jjjjjjjjj Jjjj Jvjjj Jj Jjjjjj Jjjjjjjj J Jjjjjjjjjj J J Jj Jjjjj Jj Jj Jj Jjjj Ij Jjjjijj Ijj Jjjj J I J J Jjj J Jj Jjjj J J Jj J Jjjjjjjj Jj Jjjjj Ijjj Jj Jjjjjjjj I J Ij Jjjj I Jjjjjjjjjjijj Jjjjjj Jjjjbkj Jjjjjjvjjjjjjjj J Jjjj J Jj Jjjjjjjjjjj Hjjjjjj Jjjjjj Jjjjjjjj Jj Jj J J J J Jjjjjjjj jjj Jj J J Jj Jjjj J J Jjjjjjjjj J Jjjijjjj Jjjjjjjj Jjjjjijjjjjjjj Jjj J Jjjjjj Jjj Jjjjj Jjjj Jj Jjjj Jjjjjj Jjjjj Jj Jjjjjj Jjjj Jjjj J J Jj Jjjjjjjjjjijjjjjjj Jj Jjjjjj jjjjjj Jj Jj Jj Jjjjj Jjjjjjjjjji Jjj J Jjjjjj Ij Jjjjjj Jjj Jjjjj Jj Jjjjj J Jjjjjj J Jjj Jjjjjjjjjjj J J Jj Jji Jjj J Jjjjjjj Jjj Jjj

@rachellee9998 6 лет назад

"I'm 83, I age really well. Strick diet of quinoa and steroids. I'm joking don't take steroids they do weird things. Ask my uncle he was Mr. America." --Professor Leonard what a golden quote...No way is your uncle Gary Leonard!!! LOLOL

@ProfessorLeonard 6 лет назад

Hahaha, love to have fun!! But yes, Gary Leonard really is my uncle

@dagkouta986 5 лет назад

the fact that we use the same textbook is sooo helpful. For anyone wondering, it's Calculus Late Transcendentals 10th edition by H.Anton, I.Bevens and S.Davis.

@sterlingmillhollon2520 Год назад

It is also following closely the "Calculus" 10th edition by Ron Larson and Bruce Edwards

@evanjones9602 Год назад

Early Transcendentals 8th edition too

@protocolwonder4558 11 дней назад

Professor Leonard is the Textbook we must learn from You can have the textbook but if the concept is not well explained you won't enjoy the course.

@diegol4452 5 лет назад

video 33:33 long (almost 33 secs), 333K subscribers, 3 integrals, 3D regions, Calculus 3, I'm scared!

@adriannavarro1345 Год назад

Love this man, can’t believe I didn’t watch him until now. Would’ve been great for calc I and II

@nataliemendez9153 7 лет назад

May God bless you and your family for your amazing contribution to the world, Professor Leonard! I am eternally grateful! Thank you!

@crunchyleif2368 5 лет назад

Would you ever consider putting up some kind of mock up of your exams with their answers? I feel like your tests would be much more challenging in the right way than our teachers.

@fuadsadudin1972 3 года назад

You don't know how much you helped me,you literally kept me away from anxiety,depression,etc..thanks a lot may God makes your dream come true🙏🙏🙏

@salieudanso4668 8 лет назад

I got an AA in calculus 2 and thanks to your videos. I had a tough time while I was doing calculus 3 as calculus3 videos were not uploaded. Thanks very much for making it available. With this videos I might be thinking about retaking calculus3 to make a better grade.

@ILikeWeatherGuy 7 лет назад

professor leonard is a 4d function of mass density of education. my brain gets smarter as this video goes on.

@ILikeWeatherGuy 7 лет назад

i.e the mass density of the correct electrons in place in my neurons get denser :D

@Gabriela-zf1jy 8 лет назад

I wish my calc professor explained like this and was good looking as well. I would never miss class ever lol

@tripp3468 8 лет назад

racist

@عَدِيُّ-م3ح 3 года назад

"good looking"?? what? are you going to college to marry your teacher or to study?

@vaiterius 3 года назад

@@tripp3468 how tf

@astrar7 6 месяцев назад

?????????????/@@tripp3468

@tauceti8341 8 лет назад

Single integral - area under f(x) (2-d, x,y) *OR* mass of a straight line (1-d) (??? investigate/review) "Distance along x-axis" 14:38 Double Integral - VOLUME under f(x,y) (3-D) *OR* mass of a thin plate (plane region) -thin- flat (xy-plane) Trippe Integral - Some measurement of region under f(x,y,z) (4-D concept) *OR* Mass of plate *with* thickness (cube, or surface) 14:58 ... Tripple integrals create a region just like double and single integrals. However it is defined in 3D so it a surface of x,y,z (cone/spike). Now take the mass density of EVERY point on this surface, and it tells you the mass. 18:00 Tripple integral defined S as between two Surfaces. 24:20

@Peter_1986 4 года назад

I just use Geogebra when I graph this stuff, haha. Perfect program for 3D graphs, and you can graph curves as well if you plot them as parametric equations.

@franciscodonoso1654 6 лет назад

Damn I have been using integrals for years and never really understood the concept behind triple integrals. The way you explained how single, double and triple integrals are used at the very beginning was perfect.

@mzboop3483 6 лет назад

Can't believe the difference between you and the lectures provided to me through ASU. Theyll spend 20 minutes covering a topic you spend 4+ hours on. I wish I had found your lectures sooner! Thanks so much

@liantiu8185 5 месяцев назад

Z-simple - 30:00 Ex 1 - 40:30 Ex 2 - 47:00 Ex 3 - 58:00 Ex 4 Finding mass with p(x,y,z) - 1:19:16 Y-simple. Example when you are not sure which order the upper/lower bounds are - 1:36:00 Ex 6 - 2:02:55 Ex 7 - 2:36:22 Ex 8 - 3:02:00

@justjoannak 4 года назад

Wasn't fully understanding triple integrals for my test in 2 hrs and I *NEED* to pass Calc 3 to transfer. This helped a lot, since it's a lecture

@cc-gg6cd 2 года назад

Far more informative than my math professor and actually entertaining to watch, I just learned more in this 3 hours than I have in the past 6 months of calc, Thank you!

@Festus2022 2 года назад

Dr. Leonard, Thanks so much for the great videos. What happened to lecture 14.5. Apparently, this a key video for understanding 15.6, but I can't find it in the playlist. PLEASE ADVISE.

@beatrix2745 4 года назад

I drank water right before he shouted "'CAUSE I LOVE IT" and almost spat on the screen lmaoo

@ahtishamkhan7774 4 года назад

Oh😅

@guitarttimman 4 года назад

You have a dirty mind LOL

@bryceforrest210 3 года назад

Drink water when Leonard drinks water for long life and good health

@ryanwalker4491 6 лет назад

Anyone else feel like Frodo going to Mordor while watching this? I know I do. I'm going to cast calculus into the fiery depths of Mount Doom

@charlesfkessler 3 года назад

At the beginning of the video (around the 6:50 mark) , he's talking about the ambiguity of a 4-d equivalent of volume. I wonder if you could express it as the change in displacement experienced by free space over a given time period. Maybe that's way out in left field, but it kinda works in my head.

@tiffanypham4036 7 лет назад

I would appreciate if someone can explain the problem at 2:21:56. Why the region bounded by z=0 and x=0, while x>=0 is the region in 1st quadrant but not possible for 4th quadrant? Thank you

@williammendez5209 6 лет назад

That's exactly what I was thinking.

@amjad6361 6 лет назад

i dont understand this point too :(

@eleftheriostzima5565 5 лет назад

When he defines the y-simple, he assumes: 0

@CarlosTorres290 7 лет назад

professor Leonard why did you skip 14.5 ?? also if you have any practice test up to practice !! also thank you for your cal 1 and 2 videos !!! you are my math hero !

@percycebo50 7 лет назад

What a coincidence cause I'm also writing it on this week. 13.9-14.8

@ProfessorLeonard 7 лет назад

Not to fear.... Surface Area is covered more generally, and more clearly, in a later section.

@Eunoialagom 7 лет назад

Professor Leonard professor is there any practice questions pdf sets or some book you'd recommend for practice?

@CluckYou21 4 года назад

I laughed when he said "Y, no I mean por qué". Hahaha.

@johnholme783 2 года назад

I call it hyper volume because it is a measurement of volume to a four dimensional being!

@Pete-Prolly 4 года назад

Thank you!!!!!! I've been yelling at my book for 3days and asking it "how many dimensions am I in???!!!!!" (3 or 4????) this helped a lot!!

@niquemo18 4 года назад

I want to say, thank you for this. During this time it is extremely hard to find help. This help me to the core. Z-simple. I have a great insight on integrals. I wish my professor was like you. You're amazing and handsome too.

@epistemophilicmetalhead9454 Год назад

integral is mass only if function is mass density 1st: define 3D region between 2 surfaces just like how you can switch between x & y while choosing the outer variable, you can do so with x,y&z here. there are 6 possible ways the bounds of the innermost variable will be functions of the other two variables. the second variable's bounds will be functions of the outermost/independent variable 1:03:21

@deepaksree6 7 лет назад

you are the best calculus teacher ever!!!!Thank you prof. Leonard

@wolfumz 2 года назад

2:52:40 professor Leonard says his uncle is world famous body builder Gary Leonard! What the heck!!

@terryb6802 2 года назад

holy shit

@Joeypearkew 2 года назад

You could do this but completely incorrect and I’d still not know

@debendragurung3033 6 лет назад

i am glad that i can follow him through his lecture with less hiccups. Hope he releases videos on diff eq, the troubling partial deq and calc if variations..

@tray0841 2 года назад

I hope this lecture helps me later in my education because I fell asleep watching a video about the backrooms and ended up here when I woke up

@Luicocomaro 3 года назад

theres a super sniffly student and its bothering me more than it should

@sujitsuram5075 4 года назад

57:00 1:23:00 1:36:11 2:03:14 2:36:45

@chrisrabe655 2 года назад

Awesome section. I can honestly say, I would have struggled with Calc 2 and 3 without these videos. Years later still helping students.

@BRANDONTANG27 24 дня назад

0:00 Explanation: Difference between single, double and triple integrals 40:20 Example 1 (already set up) 46:40 Example 2 (already set up) 57:00 Example 3 (setting up from given info) 1:18:23 Example 4 1:36:15 Example 5 (triple integrals with polar coordinates) 2:03:40 Example 6 (using z-simple) 2:18:00 Example 6 (using y-simple)

@engomar98 4 года назад

Please attach exersises sheets.