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Show that the centroid of a solid right circular cone is one-fourth of the way from the base to the vertex. (In general, the centroid of a solid cone or pyramid is one-fourth of the way from the centroid of the base to the vertex.)

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Johns Hopkins University

Missouri State University

Oregon State University

University of Michigan - Ann Arbor

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.

12:31

Centroid of solid cone Sho…

11:23

Show that for a regular py…

17:49

Show that the centroid of …

20:39

Find the centroid of the l…

that's who helped him. Cool. That has, um Yeah. So circle our base. So I was like, these x y c in a circle Took over these, eh? And then some height there. So you like the computer? What is this century important Issue? The century. So, uh, this is, uh, seem natural with respectable patients about their cylindrical coordinates. Uh, so because by the symmetry, X bar, we'll be able to Why? Barnum Museum. So the decentralize along the x axis So our undersea axes and see us, huh? Was the following relationship. We are. So you have here room Z her age of the morning e r. So the relationship is that See, people toe age? No, it's wormed minus. Still better. Do plugging. Um Okay, uh, equals, who are you? Get zero. So it's not be there. Are over. So these there who is seeing the height better. So, uh, for this and trivia, let's compute What is the volume? So the ruling will be internal from zero bye into gold firms. All these should buy that stuff there. No, you are from zero. Okay, run the sea eagles from zero. Cool. Yeah. So I have to There's one minor roles are over on a world that is the sea and the volume element is our first integrating Well, anything, really. So you'll get by being Joe with respect to see where we've us. Oh, just, uh is that doctor he goes doing in D. C. So that you just get tea and involvement in that. There, you get that number. The drop bombs going off age There's one minus are over A that. I'm sorry. The car so that we have in Joe really drove off are in the journal of our square Our square over ace off. We'll be sensible are in bad square. So, uh, your teen girl people too r squared hubs minus are cute. Thirds were a on these weakened zero on a So that is gonna be to buy. And sage Tim's a squared house minus in two facility. So these councils gives is on our to to these real people too, from two by age. I'm saying squirt times. So times one, huh? Online, this 1/3 cable bill. Well, my dad, my dad three minus stole that he's won over. Do with him. See? So, have you done divided by two times three. This is for you on Dean for see bar to divide, you would need to D'oh! You see, General sued by cheer up to a zero off age. It was the one miners are over A This really isn't a great over the region, O c. And then divide with volume. So, Tony, you do first the central into our Rosie Easy, Larisa. Sea squirt hopes instead of, uh, we're gonna have Well, we're gonna have that on also. You've been great. Data two by 22 by allies to buy. So go to buy items they have in there. So? So these will be able to buy because they drove from zero to a and then these fuckers squirt h squared is one Miners are over a scored thems are Are you? Yeah. So when we do that, it's clear we get one minor stew. Stems are we're Hey, my notes already. Manners are over. E squared is, uh, councils and minus. And get our square or a squared. And so you have multiplied these Why are you are r squared on our cube there? All right. Cute. So these Thea Yeah. So we are integrating these for? You know, a doctor by Sam's age squared. We're doing these. We have, uh, are scored. House miners to first are cute over a my plus you are to the fourth power doing before I'm c Square. Answer. You want to the following this year. Our name. You know the vehicle to these He's gonna turn into the square house. Minor screw thirds. Hey, you over. Hey, plus four somebody's life for or a squared so that you get a doctor. Okay, schoolgirl, these kinds of some get this squared so these people do a squared times Wow. Minus 2/3 plus 1/4. So this is equal to, um, you know, common factor off free, over floor. You want me todo so that it's got a squared. Do this. 1st 1 you need to multiply like two are 33 funding the 71 mutual reply without life or minus two times for on these when we need to want to buy. Why, three plus three. So is that over three times four. Um, so this is gonna be the six my nose heat. So these rightness school minors to last three. You saw one one so it will be a squared over won over three times four. So, what are these whole thing here? We'll be able to buy some sage square stems back them see squared over three times. Four better for ze wire. We need to divide this quantity Hide in volume so that we do that. Yeah. Bye. A j squared a squared. Yeah. Divided by between comes for I'm the volume. You were right about me, which is okay. The volume he is. Where is the volume volume is? Um the volume is equal. D'oh Bye. Each Ansari x squared my age Don't say squared. It wasn't my sweet is that he's number. So we divide those tools and cancel that. That's a long part of age Pencil. The squared. That's 30. I mean, sort of, You know how buddies at age forts. So the central it will be, but our fourth will be behind. So I come out there. You gotta the total height. Age faults, Sandra

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