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Let E be the solid inside the sphere 22 + y2 + 22 = 4 in the first octant. Then JE 2 dV is...

Question

Let E be the solid inside the sphere 22 + y2 + 22 = 4 in the first octant. Then JE 2 dV is

Let E be the solid inside the sphere 22 + y2 + 22 = 4 in the first octant. Then JE 2 dV is



Answers

Use spherical coordinates to find the centroid of the solid. The solid in the first octant bounded by the coordinate planes and the sphere $x^{2}+y^{2}+z^{2}=a^{2}$

This question. We have three ports. First, Bardem's E will be. We will have tripled integration off ex wise that these at U Y E X and the integration limit. Zero fruit four minus X squared minus y square and zero route four minus x a square and zero onto. If we take the integration, we will have double integration. Oh, x y talking like board minus X squared minus y squared. Do you want the X you choose? Equals 1/8 Integration off x four minus extra splash on forced to Theo X and it will be 1 4/3 in case toe at sort of kids. Be that triple integration off course. Three. That sign Sita Design Sita. These at the R D Sita Integration limits zero Route four minus are square zero to the little boy over to. By taking the first integration, we will have off four or bore three minus our bar. Five. Sign Zita co sign Sita. You are these either. By taking another integration, we'll have 8/3 kind Sita co sign Sita the feet up, which is equal for over three. And in a third case, K. C. That Tribble Integration Off rope Our five sign Borske Reef boy Who signed boy Sign cedar We'll sign. Seat up the euro to define the seat. The original limits zero to zero by over toe zero by over two. You have 32 over three. Sign our city, boy. We'll sign ploy Fine Cedar design cedar the fi be seater. If we take another integration, we will get 8/3 invigoration off sign. See that design Sita? These eater, which is equal for over three.

His question be you call trigger integration or through square sign Coy. Do you know B boy di Sita? Los trouble integration Off the square sign boy, you know defying BCT Integration limits zero p two p boy over three by worst You in two boy zero e Sick boy zero or worse to be zeal Boy, we're taking the first integration was science. We can get 1/3. A poor city sick or city phone de Foix de cita plus 8/3 keyboards. Three. Signed by Devoid the Sita By taking another integration, we will have 1/2. A divorce. Three regulation off casita loss for over three. A horse. Three. Integration off the seat which will be equal 11 boy he Porcari oversee

Okay, So this problem wants you to evaluate the triple integral of exposed wide Losey over the region off E. Where is is a solid that is, in the first Occident and is below the pair up. And it is below the probable oId Z equals four minus x squared, plus y the evil four minus x four miles y squared. All right, so the first thing we do is to convert everything into cylindrical coordinates. So if we were to convert our into grand into political coordinates, we would have extra swipe a Z where X is our co sign down our coastline. Dana, we have Why is our side data our scientist? No. And we have the just the and D V in cylindrical cornices are times are times DZ, pr di fada. All right, so now let's look at the boundaries. Yeah, in the first. Often that means our X y and Z components are all positive. Traveling really mattering cornets and our problem oId z before my eyes expire, Biswas, where will that can be rewritten as e equals two or minus r squared? And this is a downward facing problem. So looking at our boundaries versus DZ. So the bottom of our Z is the X Y is the X Y plane because I remember one of our arm. It is in the first octave. So x Y plane is The downward plane is a plane of Z equals zero Essentially, that restricts the region. Then we have the downward facing parable oId Ziegel. Four mice are squared, which is the upward restriction. So we could just let in before my eyes are square. Yes, we have the are So er and this obviously this largest one c equals zero because is done, we're basing problem. So we just said 0 to 4 minus r squared we can get our off to So our radius goes from 0 to 2 and finally our angle theta d'hara. It goes from zero two pi house. Now why is it pie house? Well, it's pie has because is in the first oxygen. So when the X y plane, the X and Y components have to be positive. All right, so now that we've got everything is little coordinates and we've got our boundaries now it's time to calculate it. Okay, so let's first integrate this with respect to z So our boundaries or the R and D d and I say the same. So if we integrated with respect Dizzy Well, I knew it. First get R squared the equal sign data our squares equals and data plus our square and the science data whispers he signed via was 1/2 rz sward 1/2 our a z squared And the boundaries are, of course, from four minus R squared from 0 to 4 minus are square and the other two or d r d Fate up. Okay, Next if we plot If we were to plug in the points for Mayes R squared and zero into our into ground would get the double into grown off double integral of R squared times four minus R squared. Course I do. You know waas R squared. I was four minus r squared. Sorry, they are waas. Well, it's 1/2 are times four minus R squared squared DRD data. Okay, so now we have to integrate. Now we have to integrate this with respect to D I or d thing up. Well, looking at this equation, you can probably tell that this would be This will be a struggle to integrate with respect to our So how about we try integrating it with respect to defeat data first? So let's swap the boundaries to detail. They are because if we did, with respect to D data and, er we're bound to get since we have a co sign and a sign when we're dealing with a region boundary a pirate, too. We're about to get a zero somewhere, so we'll probably end up most nicer than if we just did this ugly Integral will just back to our So if we were to integrate this with respect to data would have an integral from 0 to 2 to the boundaries of our we would get into grown r squared. I was four miners r squared. I'm silent. Data minus R squared times four minus R squared coastline data plus 1/2 one house are times for monies. R squared squared times stayed up and this is from pi over two Priore from zero to Pi Overton Oh, respected data with respect to a d. R. So now if we were to plug in our boundaries, uh, from zero department tomb, we would end up with an integral in the winter grown off of our square Four minutes or square four miners are swear plus pi over four times are times four minus R squared, squared, plus are square All right, four minus R squared, squared because off the coastline so would be at this. Oh, yeah, and respect the d r When were you just being combined? These two terms. And so we were to combine those term we could get a simpler into grow from 2 to 00 to our square I was four minus are square plus pi Ford's Our times four minus R squared, squared Respect the d r. All right. And if we were to expand this out so we can further simple fire we get an integral you go from two from zero to off Oh, yeah and be distributed to r squared into parenthesis east into our before my eyes are square Viet eight R squared When it's to our to the fourth to our to fourth plus pi over four our times 16 are minus eight R Q plus our bit to the fifth power Do you are? And now that we have this, we can integrate with respect to our to get to get in. Over three are cute. Minus to are to the fifth over. Five plus hi over four times IMEs times eight 18 r Squared. When is to argue before? Plus our six over six from 2 to 0. Okay. Wait. I see my hair here. This are not supposed to be there. I already distributed the are. So if now if we're simply plug in the values from zero to, we get a final answer of woman 28 over 15 plus a pi over three. And this is your final answer.

In this case, we have export sleep well off and why bore equals zero from the senator of the region. So the volume Equal Tribune integration off these n d Y. The X and integration limits for toe one negative 1101 And even this integration will be good for over three. So been born in this case, thin board is equal. So that board, in this case equal one Overbey trump integration off region G that the V, which is equal city over four, multiply for over five, which is equal city over five. So this Synthroid in this case will be equal. Half zero and 3/4 C five. Thank you.


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