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Find the centroid of the region cut from the solid ball $r^{2}+z^{2} \leq 1$ by the half-planes $\theta=-\pi / 3, r \geq 0,$ and $\theta=\pi / 3$ $r \geq 0$.

0:00

Carlos P.

10:26

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

Find the centroid of the r…

13:50

Centroid Find the centroid…

01:12

Find the centroid of the s…

14:39

Find the centroid of the p…

to have a Yeah, these region in three dimensional space X she and why these region c inside of our instead of us fear of these ones were these very nice is fear, right? So already is oneness Fear, Um that can be described us r squared plus c square there's that one and see in trickle coordinates Onda We got it through. The plays by even by are very well suit by thirds, uh, or Florida. That's one of your number. Wake were here to rob like these one This piece. So between that angle fables tool minus by fields to have, like Origen like these with these portion Um, so, uh, that region So for these vision de But he's, uh, instead of that for date, huh? Getting minus by thirds by thirds. I would like to find this century Sandra. So, uh oh, since I live in this region, you know, the symmetries that who said z domain and see the region isn t the same. And also for why it is symmetric with respect. Why also? And you, Sam the why minus Why? If you reflected through the ex seaplane lying those miners Why the region it stays the same. So those are the symmetries. See entries of these off the So By that reason, um, for this century are off one point. It's not for why it has to be. Zero for C has to be zero in Cartesian coordinates. Carty Shin. So you need to find out. What is that, uh, body for eggs on that? It's not where a sentry will be equal to the average body of picks which can be computed us one over the volume off that region and then integrate doing over the region. The value of X e x t y. You see. So, um well, let's try to home. Being cylindrical in in a spherical to is phrase originally spherical coordinates. Very cool. This region is described by ball data is between my nose by third's by thirds. Um, she, uh he goes from syrah off to buy the regular a range of being our rovers from Syria Off the warm is this is our unit. Yet it's here. So so That poor devil. You the volume with people toe Well, we do it from my nose. My herds upto by thirds, do it from zero by being from zero up to one developmental in this world Square. Sign off me, Sanchez, the row that furrow for feet. And then he said So integrating roll Square is gonna leave us? Uh, no Cube third's so that a volunteer between zero and one would be 1/3. Um, I didn't have control. Oh, I hope. Sign eating. Symmetrical. Minus course fee. Dari motive. Bye, Miss. You All right? Would be miners so caused by my name's cause off Syria. She's a minus eso course by his miner's wine. Of course of yours. Mine has one toe. My US magazine, with already my stools mind is like a stew. Get to to be left with the times two times dangerous from minus by thirds. I want to buy thirds. Oh, do theater from the Seadrill. Just, uh, all that baggage Elise Feta by thirds by thirds. But we have to move away meeting by thirds and by thirds to meet by thirds minus minus y Thirds said that times 2/3. And so these were the wise. So to times two, the square by over three, three times three. So our casework so very civil you I'm doing for these. Andrew, Mom for D C. Drill eggs from spherical coordinates accessible to, uh, row. Um, sign she, of course. Yeah, sort of. We would be doing. We have the intro. Well, that x road saying you course. Then devoting Parliament square sand key You rode. Okay, be set up. None. The rovers from Syria upto one the evils from zero to buy material from minus by thirds to buy Certs eso. For these four, we have Vidro First the adrenal world Cube Tell liberal, cute hero some people to roll to the fourth power number four from the line where four is your on one You just one fourth minus you. So it's gonna be fourth. Did you troll able to want for? Does he drop from my display thirds? Life turns off course we do. We have sign Feed them Selfie sile fee squared. Who feel g go. So movie drove so he scores about one off. Well, one minus course to see Rhonda into these he drove a sequel to That's We need to write that for keeping away this. The part of depends on there. So the trial of these with the 1/2 off on the job Linus causes, uh, minus sign off to free. Very right to So did that. Uh, I mean by the zero to buy zero. So this is gonna be go to 1/2 off by my no sign of buying hubs, not minus zero. My no sign of CEO hopes. So are they realize these signed off by zero. Um, sign Algieri zero. You're so you'd only have noticeable dual by house tribuna my house out of that. So will be by house stems. Warren fourth hasn drove home minus by thirds upto apply third's course theater. Be fair, but, um, vidroll off course called science for the sign. So we sign off Ferre following a meeting by thirds a minus bi third's ERM So these of Eagle Jill shine by thirds minus sign of mines by third's. Um, so if you look at our you'd circle for me half year. The sango is one be settled there. He's, uh I heard's from So this sign will be this length there. This is by thirds, is with the sign off body thirds, um, jiggle to, uh, school. It'll free or two. That is signed off by thirds. And then saying off minus by thirds. Really, Dad. So it's gonna be that party diverted there. So sign of minus 5/3 these minus scores of free hotel because you will be this protection, but it's minor that so these quantity here is gonna be squared off free over to minus minus. Squared off three or do which is, uh so that s l plus minus minus. So two times screwed the three over doing nice two rows away so that these quantities are by times the square little free divided by two. That was four. So that is, uh, missing travel. The central here earned for for the numbers while your fix That is what you wanted to compute. You have to be bite. Not in general by William. Do we have here? There were a musical to death. This is going to get so that the average eggs, what he's doing, the X coordinate for our sentry will be able to. So the volume is Oh, boys, too square. That was by We're flee squared. I mean, on this quantity, the these interval Is he going toe that ensure that turns out to be equal to did that so you will be that number by school of free, over two times drawer divided by the volume. Critical to these on these division is photo. These number that was that is equal to by screwed off. Three them. So three squared over. So to comes for And, uh, those were them's by So these, uh, by councils and have so squared off free time, so mine over. Uh, for these, with you for hems. You're four times four times two for his number 16 will be told I'm 16. Very, sir. 32 people till oh, so nine. I'm just quoting 3/30 tells and that to be Yeah, ivory starting for X, which is gonna be there. The X coordinate for a century or century would be nine. Screw the free over fairly. To that is the X coordinate the y coordinate the zero on the secret and reserve. That is it. Centrally, centrally, the war. These, uh, on that ship would be this in constitution coordinates

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