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Express the moment of inertia $I_{z}$ of the solid hemisphere $x^{2}+y^{2}+z^{2} \leq 1, z \geq 0,$ as an iterated integral in (a) cylindrical and (b) spherical coordinates. Then (c) find $I_{z}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

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Harvey Mudd College

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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.

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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.

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Express the moment of iner…

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Let $D$ be the solid hemis…

04:41

Find the moment of inertia…

All right, so, you would like to express the moment energy of the about the c axis axis of the upper hemisphere, hemisphere where the sphere is x, squared plus y square plus c square is less than or equal to 1, and so the so as the condition That c has to be greater than or equal to 0. We have here our sphere x y c, so it should be something like this: an upper hemisphere on an hemisphere, the upper side until all the momentum energies, integrating this distance over distance axis of a distance. Long distance there, the radio this distance over or the plain to the axis, so we were doing the differential element of volume and then integrating that our square over this region d so well that distance or is equal to just. We have here some point on how this so thy are with the resistance which gonna vehicle to ho times sine f, so that we have to integrate that squared, so that the interval should be of the volume element is a row squared times sine multiplied by a Square so rhoan sine squared and then so rho goes from 0 to 1. The era de goes from 0 to this angle, is by halves by halves from 0 to by halves on the inner or a missing all the way around. So that goes from 0 to pi, so this will be the moment of inertia which can be simplified to integrate rho, so square square 4 and then 1 power and then 2 powers, so 3 powers of sign off. So that is what this should integrate. If we want to look at this in a moment of inertia in sindri, so they get up here x is just a r. The resistance there is just the variable ran. The volume element is r, so this is the volume element, r d, r d c d, the, and so we would meet to integrate so r squared for the mute inertia and then or what is a tricky is the bounce. So we first orders pericles all the way around from 0 to pi. So this is a symmetric, theta, just rotate all the way around and then well. Z goes from 0 from 0 up to 1 because we're in a sphere of radius 1, so that the thisesgiven by this condition and the inner r has to be positive. So from 0, with the lower upper bound for for r is that we move bat over there. So ye'll have 1 minus c square hastur square so take square root. So this is the upper bound for r square 1 minus c square. So this will be the moment of inertia of the sphere at the c axis, because, and so all of these ones, sesthetic basis to do so were integrating first here, our cube, dr, which has sent relative r to the 4 divided by 4, so that the central Will be equal to the integral from 0 to pi from 0 up to 1 of this evaluated there so square root of 1, minus c squared over 4 vitasti, the 4 power and then minus 0, so minus 0 that is gone. I 0 multiplies 0 and minus 0, then d, c v, and so, while integrating this, this number is equal to 1 minus c square, because that is that is, like 1 minus c to the 4 halves, so that 4 halves is equal to 2 point. So 1 minus c square and so all will have 1 minus c squared times 1 minus c square which will be equal to so. This is 1 minus 2 times c, squared plus 4 for but divided by 4 point. So you have this integral. I draw from 0 up to 1 of this. We have to this at so this integral can beat them as 1 minus 2 c square, plus c to the 4 power over 4 point. So this integral with respect to z between 1 right. That is what we have there. So all this interval into 4- but that's just the this- is minus, so the interval c square is c to the third power divided by 3. We have a 2 and 4, so no 2 and 4, so it would be equal to that and then plus into tatic to the fifth power divided by 5. We have another for there, so it is that between voluted between 0 and 1, so that this becomes 14 minus 2 over 3 times 4. That councils, the that 1 over 3 times 2 plus 15 times 4 and all this minus this evaluated at 0, is all going to be 0000, so you'll have that that is equal to or magia everything was of the same exponent. We should have let that to there and then we should multiply these 1 by 3 and then this 1 by by times 3 times 3, so so that this we're going to have 15 times 3 minus. And we need to multiply these 1 by 5 by minus 102 times 5 plus 3 and that over 5 ms 4 times 3 into this is this is equal to 5 plus 35 plus 3, that is 8 over 5 times 4 times 3 and for 8 is 4 times 2, so that this whole thing is equal to 2 over 5 times 3, and while this number was equal to that, but we're left with the integral from 0. So we needed to enterie that beteen 0 multiply. We the do moment of inertia which is going to give us 2 pi there, so this should be equal to 4 pi over 15 point. That should be litemoment of inertia of this here about the c axis. It'S gonna be 4 pi over 15, assuming or unit mas.

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