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A solid of constant density is bounded below by the plane $z=0,$ above by the cone $z=r, r \geq 0,$ and on the sides by the cylinder $r=1 .$ Find the center of mass.

$\frac{3}{8}$

10:35

Carlos P.

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Johns Hopkins University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

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.

08:15

Center of mass A solid of …

00:56

Find the center of gravity…

19:34

Variable density A solid i…

06:59

01:01

Find the center of mass of…

02:58

Use cylindrical coordinate…

very nice. Solely sorry. Off, uh, constant density. Constant Lisicky on. Uh, it is bounded by the plane c equals zero the year x y z The plane Sequels Here is the X Y plane. He responded below, Like that plane on the top by the cone Our physical to see So your piece round comb Are these people to see, um on duh on the sites by the ceiling There, over his one about the C axis. So you have these, uh, very nice volume it is then interior off these so something that the I do want to find center of Must Santer off must so well on center of Masisa the average well, uh over each hornet is gonna be They are very self eggs. We are so the original Mix the aberration. Why garish z? So that's gonna be the center of loss. Our or this is symmetric with respect. The rotations. So, Fada So that these two guys, they I'm Irish Exxon over his average y will be zero to the center of mosque. Most light in the along. The secret is gonna half court in it. 00 some number. Do you know I'm gonna be the center of mosque you need to find what is. That's not That's not is a average body. You'll see. Would she have been computed of some the general over? Deal off the value off. See X Y Z over from the volume off the also first books compute. Why is that what he's got? So I think this region is, uh, in cylindrical coordinates. See you, Rico. The volume is gonna be well. We do a par, is he our? So, as you can see him, destroying on a boat from from zero up to are is the line. See, People are so ce goes from zero up to are but you can not this year are fire goes up to one the bodies one So are from 0 to 1. I'm Zoraida. He said that I would go all the way around so 0 to 2 pi So this would be the volume and then a Or be doing that your first internal you see it is your honor are just gonna you'll see polluted our so it's gonna be are zero, you know factor Ardor? No, not that. Into lives in the vehicle to our them sarsalari square er from 0 to 1. De from 0 to 2 pi. Um, so the drill a bar square? Yeah, bars Courtyard does it B r q 2/3. So that of all of it is eating 10 he's gonna be 1/3 zero. So you have that is gonna people to want third over, control from zero get to buy better. So, uh, or here that would just be Ferre Bolliger to buy my zero. So I would be I used to buy So these Morley was gonna be able to to buy third's. What is the volume? He's one. So, um, over the central enough, everyone is the same with you. Have toe lugging their a C She would have been in trouble. See, that's our You see we are Do you think on the same bones Segal's from job to our higher goes from 0 to 1. Ferre from zero to buy. It was a tradition to go. Now we're into reading, seeing this year she's gonna have won t Here are C square hubs so that is severely did that begins here on Are things gonna be our square house in my cereal So we're gonna up Dana on our square house. So we have all right square hops. Times are we are not there until about your Argos from job to one, from zero off to buy, and so over certain doing that would be integrating well, so are cute halves he are. So the intro of these have the you are to the fourth later. But for now we have ah to there. So it would be that what I didn't serum on one. So you nothing. One we're through and score my cereals of this would be one age. So I think that that was one a all metro from zero to buy this intro just there to buy. And zero said you got to play sort of. These is equal to how that too by your body by eight. So you have that This is 25 I ever ate Sudeep average. Look. See? You know, equal to Dubai divided my age and that divided by they tell volume that is equal tools Dubai thirds to by third's We give these feasible to buy. I have three delighted by a tends to buy, so to buy names. Three like a past two, but she and very nice. He's a people to, uh, three eights. So that's gonna be the center of mass with these this, uh, figure out like that she aids. So? So for that, I want to be in. They see gold in it off the center of months. Javi creates set up center of muss. Smooth blast is gonna be zero zero. You're for extra over white and then three seats for Z is in partition. Cartesian coordinates 003 It's

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