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Evaluate Ix +y +2 dV where E is the ball with center the origin and radius that lies in the first octant using spherical coordinates. marks)...

Question

Evaluate Ix +y +2 dV where E is the ball with center the origin and radius that lies in the first octant using spherical coordinates. marks)

Evaluate Ix +y +2 dV where E is the ball with center the origin and radius that lies in the first octant using spherical coordinates. marks)



Answers

Use spherical coordinates.

Evaluate $ \iiint_B (x^2 + y^2 + z^2)^2\ dV $, where $ B $ is the ball with center the origin and radius 5.

This question asks us to solve for the volume of the sphere with a radius of five. And so do that. We first have to take our rectangular coordinate and switch it into spherical so that we can compute this integral. Well, so I'm gonna switch it. And so on the outside here, I am going to put our fi and our fi goes from zero to pi. Why? Because fi goat fi has a range of zero to pi. And because this is the sphere, it goes all the way from 0 to 5. So then we do Martha and Theta has arranged that can go from 0 to 2 pi. And this is a sphere. So once more we go from 0 to 2 pi and last we have our bro. And roast starts at zero. But our road depends on the radius or sphere. And since her sphere has a radius of five, it goes from zero, that's about 5 to 5. All right. So then we can transform the inside of our integral. So the previous inside of her integral was X squared plus y squared plus Z squared all squared. And there's a transformation, we know we know that X squared plus Y squared plus Z squared is all equal gero squared. And this is still all squared. And before we complete this integral we have to add on a road squared sign of fi this just make sure that we are in spherical coordinates when we transform from articular. And then we lastly have to put our d phi, d theta and oops our defi this was real perfect. So then we can take each component and calculated out. So, I'm going to start with our fi components here are five components are just the sine of phi. The reason we're able to do this is because our entire term is being multiplied anything in here, we're being added. We wouldn't be able to separate it out because it's all being multiplied. We can separate out our component here. So we start with our zero to pi of sign the fly defy and we can compute this in a girl. So when you're computing we have negative co sign of fi still zero to pi. And then we can plug in our bounds and we get negative cosine of pi minus negative co sign of zero. And if we simplify of it, this becomes negative negative one minus negative one, which is equal teacher. Great. So next we can do our tha tha components. Well, there's nothing in this general term that is data. But that doesn't matter. We can still compute our data integral. So we have 0 to 2 pi d. Theta because there's nothing in there. We can just bring this down. And when we compute it, we end up with data. Yeah. And it goes from 0 to 2 pi still. And if we plug in our bounds we just get to pie. Perfect. So finally we can do our row terms and there's a lot of bro, so we still have to keep our bounds 05 And this road term of rho squared all squared, that becomes road to the fourth. And if we multiply road to the fourth by rho squared, we get row to the sixth, we have to keep our D. Row. So then we can compute this integral. So we get 1/7 row to the seventh from 0 to 5. And then we can plug these in five to the seventh is 7 38,000, 125 over seven. Okay, then what we can do is we can multiply our two by our two pi by our road term. And so if we do that, we get all kind of square this off. We get two times two pi tons, 78,000, 125 over seven. So if we multiply this out, we get 312,500 pie over seven. That's our answer.

Okay. We want to integrate X. E. To the X squared plus Y squared plus Z squared over the unit ball in the first quadrant. So let's take care of that first. Okay, pretend like that's the first quadrant. Yeah. Okay. So since roe is X squared plus Y squared plus Z squared. It's starting here at zero and going out To the edge of the ball. So row is going 0-1. Okay? Um E starts here at zero and goes to pi over two because that's the first quadrant equals zero or first octet tuf equals pi over two. And then data will start here at zero and go here to pi over two. Mhm. Okay so that wasn't the hard part. Make the substitution. X's rho sine fi co sign data. E to the rho squared rho squared sine fee. D zero D V. D. Stated. Okay, so now gather up all the stuff you haven't eat, the rho squared. You have a row cubed, You have signed squared fee cause I'm data. The row defeated Saito. Okay, so since the row is not mixed in with all the other stuff, we can just go ahead and pull that integral out and work on it. Okay, so the obvious thing is to let you be rho squared and then you'll need a row to make d'you. So we need to separate this into 0 to 1. Heat the rho squared rose squared row the row. All right. So if you is rho squared and d'you is to roe deer? Oh So we needed to in there? No 1/2. Mhm. And if roe is zero U. is zero square which is still zero If roe is one You is one square which is one. So now we have 1/2 zero is still zero one is still one E. To the U. You deal. So now we have to do integration by parts. So I'm going to let W. Equal you and DV equal eat the U. D. U. So then D. W. Is D. U. And V. Is eat the you. So now we get 1/2 W. V minus the integral VW. So that's one half U. E. To the U minus E. To the U. From 0 to 1. That's one half one each of the one minus eat the one minus zero minus eat to the zero. So those canceled because they strapped and you get one half times minus minus one so one half. Okay so now we can put that in 0 to 10 to 11 half. Sine squared fee. Pro sank to to D. V. D. Theater. Okay now we're gonna have to put in an identity for this. I'm gonna put the one half out in the front here, 1- Cosine to fi over to cosign theta. Do you feed the data? So I'll put this to out in the front so now I have 1 4th here to one. Okay the integral of one with respect to fee is fee minus the integral of co sign to fi we need a two in there and a one half. Okay so that we can make the fee or do you if we let you be to fee. So we get one half interview of co sign. Is the sign to feed. Um Oh these are not 0-1. This is pi over two sorry 0 to Pi over two. So now we have 1 4th 0 to Pi or two. However 2 -1 half the sine of pi which is 0 -0 0. So now we have pie over 80 pi over two integral Cosign Theta. D. Theta. The integral of the coastline is the sign From 0 to Pi over two. That's pi over eight. Well minus zero By over eight. I get.

This question asks us to solve for the unit sphere completely within the first architect. So to start I'm going to set up my bounce so for my data I go from zero to pi over two. This is because if we were to look at the unit circle, we were to cut it into fourths. I only go from here to here completely within the 1st 4th, and that takes me from zero to pi over two. So then we move on to Our fee or feed also goes from 0 to Pi over two. This is because if we were to set up our sphere here in three D. Now we only go to this point where the within the first options, we have to keep our sphere positive, so we only go one half of the way that our fee can go because our fee only goes to pie. So we have to only stay within the positive path. So that makes us go from Pi over two, from 0 to Pi over two. It's and lastly We have our row value and a row is from 0-1. And this is because it's the unit sphere and the unit sphere has a radius of one so there we have a radius of one. And so then from here we have to solve for the inside portion of our integral. So on the inside we have an X. And r X translates. Been going to cylindrical two row sign or c. Cosign theta. But then we also have E to the X squared plus Y squared plus B squared. An X squared plus y squared plus Z squared is equal tara squared when we go into cylindrical. And finally because we're going sorry spherical, That's wonderful. And finally when we go from rectangular to spherical we have to remember to include a row squared sign. See And of course we have our D Rho D. V. D. Data on the end. Perfect. Well if we separate out each individual portion, we get From 0 to Pi over two coastline theater, the theater. And then we also have from zero to pi over two sine squared fee. The fee And last we have from 0 to 1. We're all cubed E to the rho squared D. Row. So have cleared up the screen so that we have some room to compute these into girls. And just start, I'm going to start with data So that the data integral and want to rewrite it. It is zero to pi over two coastline of theta the theater. So simply you just have to solve this integral. And this is a really basic integral. And so when we go from when we integrate co center data we get sign of data From 0 to Pi over two. And from here we just have to plug in our bounds and solve and we plug in pi over two to sign of data, we get one minus. When we plug in zero we get zero. So our answer here is one. Yeah. Great. So then we can move on to our fee And I'm going to rewrite it here, it's zero to pi over two sine squared fee. Gov. Well to start, I'm going to use a trig Identity just to make this a little bit easier. That identity is that one? That sine squared of data is equal to one half, 1- Cosine of two Data. So I'm just going to write this So it's 1- co sign of two fee. On the end we still have the fee. So great. So now we can compute this integral because this is much easier to solve. So on the outside we still have our one half, but on the inside now we have a fee minus. And when we compute this integral, no matter what we do, our sign is going to make it be equal to zero. So I'm just gonna put a zero there And then our bounds are still from 0 to Pi over two. And if we plug in our bounds we get 1/2 times pi over two, Which is equal to Pi over four. Once more, I have cleared the screen and now we can start working on row. So I'm going to bring that down. So we have Integral from 0 to 1 bro, cubed E to the rho squared. Deep bro. And so for this one, I'm going to set U. Is equal rho squared, so that do you, is equal two bro, dear. Oh And so then if we redo the integral we still have our bounds are from 0 to 1 means when you plug goes in it puts out zero and 1 again. So but then if we had separated this real cubed into row and row squared we're left with a row and we have a you and we have E. To be you D. You over. Do you wrote to grow and we have a deal bro left. So then we can cancel out some things such as D. Rose, the singular Rose and then you can write it out. We have 1/2 And a girl from 0 to 1 U. E. To the U. Do you? This is a much more manageable equation so don't do this. We need to integrate by parts so I'm gonna set her well what would be a U. Value to you and R. V. Prime as eat you. And then I'm simply going to integrate on this side. I'm going to differentiate on the side and integrate on this side. So if I differentiate I have a one and then if I do it again I have a zero and over here my eating the you just continues now and so then I'm able to connect these diagonally and I have plus and minus chilling can rewrite my integral as one half times you each of the you minus each of the you and our bounds are from 0 to 1 so that when we just we can just plug in are bound. So we have one half e minus e minus zero minus one. And if we simplify this all just equals one mm. So then if we bring in proportions from the previous equations we bring down our one, you have one Times Pi over four times one half. So the answer for our total integral is Pi over eight and that is our final answer

Were given an integral. And we're asking you spherical coordinates to evaluate it. Sorry mm. This is the triple integral. Excuse me. Over the region E. He ordered cafe con Leche, sweating out the order coffee militants. That's what we use a question. You and your boys have been hanging out. Yeah. Of X squared plus Y squared D V. Where E is the region that lies between the spheres are oh, first of all, this is steel, X squared plus Y squared plus C scrape was four. Use the voice over at sausage party, homework in X squared plus Y squared plus B squared equals nine. Yes, he is played. He played Andy's little sister. Mhm. Green. To make the change to spherical coordinates first. Right E in spherical coordinates. So because we live between these two spheres, we know that rho squared is going to lie between four and nine. So that row lies between the two radio guys, two and three of the spheres. As for phi and theta wealthy. That ranges from zero to pi still and five from zero to pi sausage. So some you hate now we can use spherical coordinates to write are integral as the integral from 0 to 2 pi integral from zero to pi Integral from 2 to 3 of our function. But now in spherical coordinates X squared plus y squared becomes rho squared sine squared phi. Okay. And then D V becomes rho squared sine phi Hero D five d. Theta sauce label. Like I feel like We can rewrite this using Fujianese theorem as a proactive integral and we get the integral from 0 to 2 pi the theater times the integral from zero to pi of sine cubed. Fi defy Times the integral from 2 to 3 of Road to the 4th. Dear. Oh, it was really a chicken nugget. Also, chinese in Wisconsin. Appleton Taking anti derivatives. We get data from data equals 0-2 pi. The anti derivative of sine cubed. If I this is a negative co sign fi these are plus co sine cubed. If I different sauce cover Syria Children's Over three. I mean felt like I was going to destroy got some real California voice from phi equals zero to pi times. Uh Yeah it's actually named Road to the 5th over five from row equals 2 to 3 shows to the next month. Your mom and your so plugging in values We get two pi times. Use italian the fuck mama. The ranking male men or thirds. Mm The ceremony. Thank you Germany. Yes. Do you? How many people fuck by the really great pleasure kissing you directly. And so yeah. Mm. Times. And this is 3 to the 5th -2 to the 5th Over five. Mhm. And solving this is equal to 1688 pie over 15. Excuse me get married. Sure candidate.


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