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Find the moment of inertia of a solid sphere of radius $a$ about a diameter. (Take $\delta=1$ )

$\frac{8 \pi}{15} a^{5}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

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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.

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.

08:22

Moment of inertia of solid…

10:57

Calculate the moment of in…

02:12

We have an atmosphere and we would like to find the moment of inertia but its diameter, so i lit if you make the rotation rotation for the center, so the movement of inertia is gonna, be to integrate or the sphere this distance are squared. So the distance to equal these xaxis, then you have here x there y so see what this is choices to do spherical, so the omit of inertia of a 1. We call this a sphere. Nothing would be to integrate over as the distance or square. We assume unit density, so this or the sphere would be the distance or square in a spherical coordinates will be rho sine of this rho sine of t square. So you need to integrate rho sine of t square. The volume of a man for he is squared sine of on the row de de there, where all goes. This is a sphere of radius. A so goes from 0 to a f goes from 0 up to pie, and they are as all the way around 0 up to 2 pi 2 pi is so o here, our first interal interroge square times, whole square plus 4 italo. These is row to the state power by 5 and we're going to a pallor, 18 and 0 will be just equal to a t- the powered by 5. So that's going to be the integral with respect to row, so we have a to the fifth power divided by 5, and so here we can, since there is no teadependence, we can can integrate. So the integral of this rather will give us 2 pi so that this would be god times 2 pi times the integral from 0 up to pi of we have sine square of times sine of p d. So that's going to turn into a sin cube of, but a sign cube is equal to sine squared times sine of p v f, and here we can see these by the 2 integrity- that's equal to 1 minus cosine squared so that if we do any substitution, If we see that, u is equal to- u is equal to cosine of t is equal to cosine of t that this is equal to u square. It was cosine of p. So, d, u will be equal to minus sine of t t this integral would be the integral of minus this part i should a minus 1 minus: u square du and the 12 bis! Minus? U? 2! Minus? U! Plus? U to the third power divided by 3! Right, so all that evaluated between- so u is equal to cosine of so to you. Think cosine of v and v is picosine of 0. So these 2 points are cosine 1 on tis 1, so that this will be equal to minus minus 1. So would be equal to this bracket, minus minus 1 plus or minus 1 cube and that minus so another minus 1 and then this 11100 to power like thee. So this number is equal to minus 2. Thirdthis is equal to plus 1 to minus 1. So this is also equal to 2. This is a 2 third, and so we have 2 thirds minus minus 2 thirds is equal to twice 2 thirds 2 times 2. Third, so that this, this integral is so we're gonna have times 2 pi divided by 5, and then this integral is that times 2 squared divided by 3, so that this is 2 square times 4 times 2, that's 88 equal to power, pi and then divided by 5 times 3 died by 15 point, so that is the that is this moment of energy of this, the sphere of it that its diameter is 8 to the fifth power times pi over 15 point.

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