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Consider the region E that is above the surface 2 = 0, inside the surface %2 + y2 = 4, and outside of the surface 2 = Vr2 +y .1. Write a a triple integral describin...

Question

Consider the region E that is above the surface 2 = 0, inside the surface %2 + y2 = 4, and outside of the surface 2 = Vr2 +y .1. Write a a triple integral describing the volume of E using Cartesian coordinates (that is, €, Y, and 2)_ Draw picture to explain: 2_ Same question in cylindrical coordinates_3 Same question in spherical coordinates.Which coordinates would you use, and why? You do not need to compute the integral_

Consider the region E that is above the surface 2 = 0, inside the surface %2 + y2 = 4, and outside of the surface 2 = Vr2 +y . 1. Write a a triple integral describing the volume of E using Cartesian coordinates (that is, €, Y, and 2)_ Draw picture to explain: 2_ Same question in cylindrical coordinates_ 3 Same question in spherical coordinates. Which coordinates would you use, and why? You do not need to compute the integral_



Answers

Use cylindrical or spherical coordinates, whichever seems more appropriate.
Evaluate $\iiint_{E} z d V,$ where $E$ lies above the paraboloid
$z=x^{2}+y^{2}$ and below the plane $z=2 y .$ Use either the
Table of Integrals (on Reference Pages $6-10 )$ or a computer
algebra system to evaluate the integral.

Were given a region E. Yes. And rescue cylindrical or spherical coordinates To evaluate the triple integral over E. Of Z. Mm hmm. Our region lies above the Carrabba Lloyd. Z equals X squared because the white square And below the plane's equals two. Why Rescues either the table of integral or a computer algebra system to evaluate this integral? Sure. You know, I'm so bad now thinking about this region. It's above a tabloid which isn't really spherical to begin with. And below a plane doesn't really appear to be part of the spear. So it makes more sense to you cylindrical coordinates. So we can describe it in cylindrical coordinates with data between zero and 2 pi as for our well, yes, we want to find where the plane and the collaborator of sex. This is when X squared plus y squared equals two. Why now in cylindrical coordinates this is the same as R squared equals two times are sign of data and therefore are equals two times the sine of beta. So well that are arranged From as small as zero up to two times the sine of data. No, it's things literally, but no, no, just just yeah, actually made a mistake here. So because our planes equals two. Why is a. Uh huh. Way it works. See it lies along the X axis that contains the X axis. Uh huh. It follows that data actually ranges from zero up to pie. And now our ranges from zero up to two times this sign of data. Cold car instincts, she said. Yeah right. Had. All right. And then find Z. Well, you know, the Z is going to be greater than or equal to expert plus Y squared which is mm Yes. Our sweater less than or equal to why? Which is to our sign of fear. Maybe a single joke. Yeah. And so our triple integral over E. Of Zdtv Becomes in cylindrical coordinates integral from 0 to Pi negro from zero 22 signed a data. Yeah. And integral from Z equals R squared to to our signed data. And Z. This is now it's nosy. And then D V becomes r DZ DRG data. This is equal to Any growth from 0 to Pi. And the girl from 0 to 2 signs data. Right? And then give me a derivative with respect to Z. I get our times Z squared over two from Z equals R squared to to our science data. He er he stayed us. This is the integral from 0 to Pi Integral from 0 to sign the data. And this is now see one half times are times four R squared sine squared data. The four our cube, sine squared data minus. Our times are squared squared. Which is our to the kids. E. R. Dictator. Always just sure This is equal to 1/2 And the integral from 0 to Pi. We all look the same. So and I'm taking the derivative with respect to our we get mm Art of the 4th. Sine squared data Minour are the 6th over six From R equals 0 to to sign of data. The data. That's serious there one half times And taking anti derivatives. This is not yet. This is any girl from 0 to Pi. Uh we should It's the art of the 4th. This is now or to the board to do this for which is 16 signed the fourth on sine squared has signed the sixth of data minus to the sixth which is 64 6 Signed to the 6th of Data. The data. Got it 32 3rd mm. Boys what? This is one a half times 16 3rd integral. From zero to pi Signed in the 6th of Faith. To be fatal. So yeah, it would. To evaluate this integral. A little bit complicated. Well, first of all we use trig. We can write sign to the 6th of data. Yes, this is the same as sine square to fade A square times sine squared of feed us. He he is trying to write this as um Mhm 1- Co signed Tuesday to over two. Just wind kick like excuse me, I'm trying to sleep. Excuse me. Times one minus co sign with two theta over to which is mhm. 18 times one minus to co sign of tooth data plus coast signs, swear to tooth data, times one minus the curve sign of tooth data. Well. And then was interested in his gun. This is 18 times one minus to co sign of tooth data plus one plus the co sign of two times two or four data over to. And one man is the co sign of tooth data. This is 116 times two -4. Co sign teeth data Plus one. This is sexually now three Plus the co sign of 4th graders. Times 1 -1 Sign of Tooth Data. All right. Oh just saying just saying oh you know this man. Hold on. Yeah. You have an award James. Mhm. Well the best rest and I then using this year more trade values. You can write this As 1/32 times 10 -15. Co sign to data. I'm just going to start going to everyone Plus six co sign of 4th data. Yes that's it- Co sign of six data. Most. Mm. You speak well argument. Yeah. So are integral is actually equal to Yeah. Eight thirds think oh Times the integral time is 132 Kind of being a girl. Mhm. From 0 to Pi of 10 minus 15. Co sign of tooth data plus six co sign 1/4 data minus the co sign of six data data and taking anti derivatives. We get said This is 8/3 time. is 132nd times and data minus 15. Classic. Mhm Sign of tooth data over to plus six Sign of 4th data over four. Which is the same as three sign of fourth data over to Yeah minus Arizona. In a sign of six data over six From data equals 0 to Pi. Yeah. And in fact this introduces 28 3rd times 1 30 seconds times 10 pies. This which reduces to five Pi over six. As the problem says, instead of doing all this tree, you can alternatively use a computer out of the system. You should get the same answer. Five pirates.

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.

So, um, oftentimes we can't just take a single integral. We often have to take double integral Zoran this case triple integral in order to find important values. In this case, we're finding volume. We also know that it's important to be able to switch between coordinates because, um, things aren't always rectangular. Oftentimes it's easier to use cylindrical or spherical coordinates. Eso What we're going to be doing is working on our ability to use those different coordinates to make problems ultimately easier. So what we have here is we first want to determine the range for theta. Um, since we know the regions in the first Occident, the angle will range from zero to pi over two. Then the Z value. We want to convert into rectangular coordinates. Since Z is equal to four minus x squared plus y squared, we will just call this four minus r squared since x minus X squared minus y plus y squared is R squared. Then we know that when Z uh, we can plug in Z equals zero. When we do that, we see that our is equal. Teoh a radius of two. So with all this in mind, we now can take are integral. We know that the fate of value is from zero to pi over two. We know that our radius is from 0 to 2 and then we have our functions E which is represented by four minus r squared. And therefore we have our X plus y plus z Easy. De are de Seda. Um what we can dio with this is, um, move the theta right here. Move the theta, right? Yeah, When we do that, we know that this is just gonna be pi over two. Similarly, we can we want to represent And before we do that actually will want to represent X, y and Z as the appropriate, um, variables. So X we know is going to be our coastline. Fada, Why will be our signed Fada and Z will just be easy, but we want to have it's going to be our DZ drd fada. So when we multiply everything through, we get our Z r squared sine data R squared coastline data evaluate this simplify, But we can also use these calculating tools because they're very useful in finding inner grows quickly s So what? We end up getting is this? And another way we could write it if we salt it out, would be 128 over 15. Class 8/3. Hi. Um, as we see these answers, check out. So we know that we did it correctly, and this is how we will conclude this problem.

This very nice sphere. A sphere that as the following equation And he says fear off radius squared off to this is in our free and we cut it by the cone. God, why they call him given by sea is ableto squaring off x squared plus y squared. So the dynamics look like the following me off here. See, X, why are they now our sphere? Something like thes uh, very cool. Oh, steel sphere, center of the to the origin on. Uh, I mean, so you have this nice sphere, and we have the This comes, uh, that cuts it through there. I've gone through consider the area of the lower in the work up, so lower cup would correspond to this portion. So the possible goes, yeah, or the you know, the Yeah, that looks like a Yeah. Like a round stop. Something, like exactly. So you have to find them the area of these. Okay, um, so what we can do? Hysterical for it. Very good coordinates. So, since the raid, this is a squared off to exceptionally given by spirit off to and, um, sign she cause Vera, why is he going toe squared off? to and sign off fee. I'm a sign of era and C is ableto squared off two times. Um, of course, cause I know she so deism. Uh, these, uh, now Ah, yeah. So the angles, um, fiance are like the following scene you have here a better were there with distance. All the distance here is squared off to So the angle the but the spectrum makes with the C axis they see is here the X the Why decide on this field on, you know, over under the projection on the X Do the X Y plane makes an angle with exactly so the angles there. So that, uh, for this region, um, by the symmetry, uh, off goes all the way around, as you can see, so that I will go from 0 to 2 pi. Uh, there are other agencies squared off to on the on the annual fee. Let's libera and that that piece about area sideways. So we have here the sphere and we look sideways Looks something like this. So these angle here is about yeah, equals two pi for so that to range from there Off the over there. There is there the lower part fairness. Now he needs to go from 5/4 up to buy. So our in general would be controlled from five forks off by and, uh, from 0 to 2. Pi off. Mom, all the differential. The differential ovary. A better. It was gonna be too. Um, sign the sea. And then we have the P. Yeah. Yeah. This differential of area is equal to, um that is going toe our Silveira cross brother with are soon data crossbred with RCMP that, uh, all the normal bob on this cross product can be computed us the determinant of this matrix. I j Okay. And then where these role be right, Arsene Feta. Another God Final roll RCP. So these are the I think some of the components of the respective era. So the active off extreme respective failure would have minus scraped off to Tom's son fee on, then sign Keira. We have developed X with respect to Forever in Iraq. Caused so much so it's called off to sci fi off course. Terra, They won't see with respect to theta zero and then not a big city inspector. He will get cut off too. Thanks, cause three sub cause Karen. Yeah, And then for a white for what we're gonna have screwed up to because he if Sanofi sign. Okay, uh, on the attic off. See, with respective fee would be minus is cool it off to I'm sign. Yeah, child field. So this vector has components. Items distance that. So items minus two. When it's too high, them sine squared off times cause yeah, on then Or J we have That means that minus that times, that's a minus. Toe sine squared, but still J sign squared B um, sign off era. Okay, we have, um, minus two my school k. Yeah. Uh, science square off era. Yeah. Uh, scientific Castro sign off key. A sign of bro on for the beginning of the game not to, but because there's a minus two minus two cause the square off era mhm, uh, courses square. Uh uh, cause fi exactly. Scores P sign off. He should have all the normal. These vector. It's Norm. Compute the normal bees with people too. So we have a this case here. Uh, mhm. Since we have our yeah, signs, we can factor a sign square plus courses square because we'll sign off. Ecos fee is multiplying balls so that we can factor that that is able to one on then peace. All of that here is just sign off feed, cause he if you so as you can see way have our you have a sign of feed coming to There were two sign of going to off those. She's gonna be all the components. This car is gonna be two square sign. She squared. Helene, this is Claire will be again. A sign off the square times called squared. Plus, uh, sign off his word sign Famous squared. And in that, that plus, uh, here we factory to sign feet. So plus cause the squared. So we need to take square it off. That for normal Mrs. Squared off this whole thing. So this is gonna be equal toe. So we have ah, over this one. We have, ah, causes court. Assign square that it's one. This will be able to move to to square square with X squared, so you'll have to sign off fee when the times the square root off. So this factor is just signed. Please squared on them, plus cause if you square okay, which is ableto one so So that the differential over is too silently. Great. Because we have said the school sign coffee so that the area is gonna be able to this. Yeah. And then, uh, bodies goes to from zero to fight Syria. The Dubai. So this integral or the jail off to sign is, uh, minus tool call. Sign on the evaluative. Getting by force on. Uh, bye on, then. All off. After that, we have to integrate. Uh huh. So that yeah, so? So it is that so with the, um, Because minus, uh to casino, Uh, 54 if you want todo or the minus 2% of five or four minus my costume co sign off by mhm and then hold at times. Um, that times these intervals intel from zero. Thank you to pile hair is integral. Is able to buy so many people to five. Mhm. And then here, uh, co sign of fire for is able to squared off to or two. And then because I'm a size minus one. So Well, so it would be, um it's You have to minus. Yeah. Yeah, on. Um, So you have these in dropping draw to sign with U minus two because, um, as we have, um, I see each daughter. So you should be casino. Bye. So the order off integration is the opposite. Because I know fight because I have fire force. It would be minus because I find minus minus minus. Because I have 54 That's so these should be ableto This is not a vehicle to minus two uncles. I applies minus one on. Then here is gonna be minus, uh, minus two times because I know five or four is screwed off to over to so that these councils and gives us square with two. So these should be equal to minus minus. So that is up to minus minus again. So two plus a squared off to okay, can stoop I, which can be read in US four plus two, squared off to by. So that is the that is the period


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