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Find the centroid of the solid bounded above by the sphere $\rho=a$ and below by the cone $\phi=\pi / 4$.

$\frac{\pi a^{4}}{8}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Johns Hopkins University

Missouri State University

Harvey Mudd College

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.

12:26

Centroid Find the centroid…

01:09

Use spherical coordinates …

11:03

Use spherical coordinates.…

00:58

Atmosphere of radius, a solid, solid here x x- i i is the i have this very nice sphere. Radian then we got through the cone described by f is equal to this. Cone is young, o p is equal to pi over 4 it so half of 10 up there is by halves, which is going to have it by halves by 4. To have this cone this shape. Let'S call that region, we have to find the centroid centroid of the so by the symmetries. This is symmetric with respect to therein the workin, the spherical coordinates, or we have real zere, so symmetry with respect to there, so that the central the average x has to be 0 and the average y has to be 0 because he doesn't marry. We rotate the figures is the same. We need to find what is there that point that is going to be the average c average c. That'S going to be c, not the z component for the centroid, which can be computed as 1 over the volume of the times, the overs z, davis, value of z inside integrating over d. So for this volume for the volume you can do, spherical coordinates. So we do the volume lumiere square sine sine of the w rho d e t t to waller this. This goes up to radius, a that is rho pisana a tin is going to be between 0 and 5 over 4. So for all we have. That goes from 0 up to a e goes from 0 up to pi over 4 and there goes all the way around to 2 pi very nice. So if you do first integral with respect to theta, we have this integral theta from 0 to pi. You can do that, since nothing here depends on the so their integral is just 2 pi, 2 pi. So we're gonna obtain a factor of 2 pi from center of from 0 to pi over 4. When we get rid of that interval, the petofi try from 0 to a over a square sine of 3 d, so the interval of row square. This will be equal to rho, cube third, and so that evaluated, withween 0 and a you will obtain 2 pi times a cube times integral from 0 up to pi over 4 of sine of 3 d, and the interval of sine of t is minus. Cos minus cos lof p, so so, if we have that to be that between over 4 and 0 is going to be equal to minus cos of pi over 4 minus cos of 0, so this cos of pi over 4 is a square root of 2. Over 2 and to 0 is 11, which can be written as 2 over 2 so that by doing this, substruction can be written us. That is minus that so that minus 22 minus square root of 2 over 2, which, if we have that minus, can be written as 2 minus square root of 2. All of that over 2. So you have that that that volume is equal to 2 pi third times a cube times: 2 minus square root of 2. Over 2 point we can cancel 12 there, so this is a that is the volume. So the volume is equal to this volume is a cube, timso minus square root of 2 over 3 and then for this this integral or we can do c- is equal to rho cost. So you'll have ro cos p and the volume element is rho square. Sine of f, so we have that then 0 t d, their row goes from 0 up to a or b goes from 0 to pi over 4 on theta from 0 up to 2 pi point. So here what we have the integral row cube how from 0 up to a this would be equal to row to the fourth divided by 4, so that evaluated between a and 0 is going to be a to the fourth power divided by 4 minus 0 point. So i'm going to think that i'm a factor of a to the 4 divided by 4 and we can do the integral with respect to 0, because there is no function depending on theother. So the there from 0 to pi. This just going to be equal to terval 2 pi minus 0 to 2 pi, and then we have that. So we go to that times. 2 pi times this integral interval left is interal with respect to that is enteralfrom 0 up to pi over 4 of cost sine f d f, and for this we can do. We can do 1 on you substitution in, let this be? U? So? U is equal to cos p. We say that this is our? U d? U would be equal to minus sine f t f, so that this integral would be minus the to of? U d? U, because you have there, u and then dusius minus, and then we have here for the bounds for what is at pi over 4 and for what is at 0 pit. So yet pi over 4 is going to be cos of pi over 4 cos pi over 4, which is equal to square root of 2 over 2, and that 0 is going to be cross of 0, which is 1 so that this integral is going to be Minus? U squared squared halves and the evaluated become square root of 2 over 2 out of 1 point so that this becomes minus square root of 2 over 2 squared and then 1 half of that minus 1 square, which is 1 divided by 2 t. So these numbers here is gonna, be dis order, gonna be 2 divided by 2 squared, so these 2 will cancel together. These will become 1 over 4. This square minus 1- and this over is equal to this- is 1 half is equal to 2 over 2 square, which is 4 point, so these would be equal to minus so minus 1. Minus 2 is minus 1 over 4, which is minus 1, for that is, for this is by 1. Fourth, so that this hoping o this is 1. Fourth, so that this whole interval will be a to the fourth power times, 2 pi divided by 4 times. So this was that integral- and this is the riding so that for c bar, we need to divide that integral by the volume, so that t bar would be equal to the verse c, which is the c component for the centroid is going to be. That number. A to 4 times 2 pi divided by 4 times 4 and that divided by the volume that the volume is equal to that divided by a cube, have 3 and the mormon 2 square root of 2. So that is the volume and then all these can be written as multiplied that by that multiplied that by that tastes, a factor of pi. So you have so it will be equal to a to the fourth times: 2 pi times 3 divided by 4 times 4 times pi times a cube times: 2 minus the square root of 2, so that this becomes this councils 3 powers there. So we get up for a 1 power of a a and then it cancels, and then we have a 3 and these councils are 2 there. It would be equal to a times pi times 3 over 2 times 4 times 4 times 2 minus square root of 2, so that this is equal to 3, a pi over 82 minus the square root of 2, and so that is going to be o. That'S gonna be the value central, the c coordinate for the century. That c not is equal to that.

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