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Find the volume of the region bounded above by the sphere x' +y + z' = 2 and below by the paraboloid z =x' +y12.2587 b.2.9743 6 18512 413456...

Question

Find the volume of the region bounded above by the sphere x' +y + z' = 2 and below by the paraboloid z =x' +y12.2587 b.2.9743 6 18512 413456

Find the volume of the region bounded above by the sphere x' +y + z' = 2 and below by the paraboloid z =x' +y 12.2587 b.2.9743 6 18512 413456



Answers

Find the volume of the region bounded above by the sphere $x^{2}+y^{2}+z^{2}=2$ and below by the paraboloid $z=x^{2}+y^{2}$

Three, huh? Our region that he's, uh you say over thes is fear. Yeah, that's fear described. Very acquisition. That, uh, excess queer. But White Square last C square wasn't to fear of Brady's square Go through from dinner. Well, uh, you can just consider day over partners. He's, uh, supposed to be over who's this region? But he some all these service that is see, people kill x square. Plus why Split? Said I have them. Oh, look, er oversee our sea will be between thes number, which is cylindrical. He's, um, r squared treating that, you know, because you go through. See, he was two miners are squared schools, so the sea will go. You think those numbers well are square square? Do minus R squared on the I am civil are to find what should be there. He's gonna be one of these two meet, but so well, that happens when these is equal to see. So you have. See, WASI Square is equal to so that if you move these to appeal your side, you know. But you can see that these factors us z my no stool them See us one, uh c plus through see my nose. One, uh, CIA has one. That's good. Right? Because if we multiply still, you might wanna be out to see mine of senior year. Uh, she and then C square, so, yeah. Ah, some of the radius will go from I was gonna be easier on one, because that is when that's he. It's their, um Well, the Martian there is one, and, uh, are fair. Right? So, er was supposed to be all the way around. So Clara is between seer And by so to compute the volume of these reason these volumes his body was gonna be so the internal from, uh well, si goes from are up to our squared off our square off groups were Oh, uh, to minus R squared. That's easy. Wherever boarding element are, Jack. Oh, yeah, Mom. Part goes from 0 to 1. Are you armed? They're also the way around to buy. So we're doing this in the room. You get just difference between those two points he's gonna be, um Well, let's see. Two great first data because but in dirty basil dedazo did you get that? We're gonna get to just having to buy times are these all from zero. The worm led our But you're doing this. Uh, our time's just Valerie conceive eating those points. So that would be square the to minus R s. Word my ass r squared on so mmm, that you are earned. Uh, well, we have two intervals until far squared. Two ones are squared. My moment. So what, We can see that these in to go these to be the u you people to two of my aunts are squared. Do you see Balto? Minus through our we are. So these would be the internal miners. But we have good bye to so Rayna's worm. He drove or you 1/2. You physical too? Well, miners. One hard on three times. 2/3? No. Then you know Well, you did a three house and in that hour, dirt into fee. Poor folks. Those two guys also, it's gonna be, um, you to this tree house. But you people, too to my nose are square said the central will be minus two miners are squared. The three house run divided by three Andi in a world Until these were tying to the drop off our cube, the other internal injuries minus the jewel of our cube. She's, um he's got me heart to the fore by four. Um, sort of, uh, well, to a decent girl, these, you know, he's gonna be to evaluate these this expression routine. G begin one as you 18. Now, one as you. So So doing this at one. We have miners do minus one. So was 1/3 armed, minus one, both on the miners, then zero you're gonna have Where? Your stupid three house. There still is a real house. Nerves. Oh, my suit be This whole thing is gonna be you. Drink three. House is you, too. Yeah. Was it did well, to have these, Uh, well, the whole thing will be to to buy. Thanks. Too screwed up to. Yeah, who's good in thirds? Really? Minus 1/3 in my class. This one 14 Oh, so he could, uh, you can. Well, fine. Fine going, denominator. Multiply life for here. So we have a full order. Multiply way for there. We have a full bear. So is it me four times do That's eight. So it would be Hey, cool. All right. My nails for his one. So minus five. Oh, but ah, you knows how to I want to play three. So it is quite true, mind. 43 she's mine. And seven, um, three times four with the other by there like that. So people buy over, uh, three times to those in six. A skirt off, too. My assessment So that but he's this should be that beautiful. These these region. Um, these by any expert to my seven or six.

Were given a region and were asked to use cylindrical coordinates to find the volume of this region. This is the region bounded below by the plane. Z equals one and above by the sphere X squared plus y squared plus C squared equals four. So first of all, we need to be able to describe this region of integration. To do this, let's figure out the curve of intersection between the plane and the sphere. What we have that X squared plus y squared plus substituting in one squared is equal to four. Using both equations and therefore X squared plus y squared equals three or in polar coordinates. R is equal to the square root of three. So this is the curve of intersection and therefore the projection onto the X Y plane. De is given by data between zero and two pi and are between zero and the square root of three. Now our region is bounded above by the sphere and below by the plane. So we have that a region M is given in cylindrical coordinates as data between zero and two pi are between zero and the square root of three and Z lies between one and while we're looking at the upper hemisphere so Z is greater than or equal to zero in solving the equation of the sphere X squared plus y squared plus Z squared equals four we get the Z is equal to the positive square root of four minus and an X squared plus y squared becomes R squared. So Z allies between one and the positive square root four minus r squared and therefore the volume of this region the is defined to be the triple integral over this region m of one which becomes we change coordinates the integral from 0 to 2 pi integral from zero to route three integral from one to the square root of four minus R squared of our function one times the differential are D C D R d fada and first integrating with respect. Excuse me. Integrating first With respect to Z, we get the integral from 0 to 2 pi integral from zero to route three and you can pull out in our times square root of four minus R squared minus one d r d theta and we can use you Beanies. The're, um right. This is a product of Integral, so we get integral from 0 to 2 pi d theta times integral from zero to route three of our times The square root of four minus R squared minus R D R Evaluating to get two pi times Taking anti derivatives Here This is the coefficient of R squared, which is negative one times one over the exponents, which is to times one over the new exponents for four minus R squared, which is three halves. This becomes two thirds times four minus r squared to the three halves, minus one half R squared, evaluated from zero to Route three and so plugging in our values. We get two pi times, so we have a negative one third time's then we have four. Minus three is one 23 halves, so negative one third minus three halves minus and then we have negative one third times four to the three halves, which is eight to negative eight thirds minus zero. So this simplifies to well, we have two pi times and negative one third, minus three halves minus negative. One third times. Eight. We have negative one third plus eight thirds is seven thirds, minus three has and this simplifies to. Let's see, we have 14 minus nine five. So we have 10 pie over six or five pi over three, and this is our answer.

Were given a region and were asked he's spherical coordinates To find the volume of this region. The region is bounded below by the plane. Z equals one and above by the sphere X squared plus y squared plus C squared. So just as in a previous exercise, we have that the curve of intersection satisfies the two equations, Z equals one and R squared plus Z squared equals four. So we have that r is equal to route three and therefore our curve of intersection. We project onto the X Y plane get our Domaine de This is given Well, okay, If we were doing cylindrical coordinates, that's how we do it. What we're doing spherical coordinates. So it's a little bit different. So first in spherical coordinates, the equations equals one becomes row times the co sign of fi equals of one and the equation X squared plus y squared plus Z squared equals four becomes row squared equals four. So that row is equal to since it's positive to and therefore we have a curve Intersection is when these two equations are satisfied. So we have the to co sign of five z equal toe one or the co sign. If I is equal to one half because fire lies between zero and pie, this implies fi is going to be equal to pi over three. So you have that a region w could be described so that fatal range from 0 to 2 pi we have that five will range from zero two pi over three and we have that row will range from well we have from mark. Equation of the plains equals one ro is equal to one over cosine phi or the seeking to fi. So rare ranges from thes seeking defy to yeah, to the radius of our sphere. And so we have using spherical coordinates the volume of a region which is the triple integral over w of one. This becomes the integral from 0 to 2 pi integral from zero to pi over three integral from seeking to fi to to of one times our new differential which for spherical coordinates is row squared Sign Phi dear Oh defy d theta now taking the anti derivative with respect to row, we get integral from 0 to 2 pi integral from zero to pi over three and then we have sign Phi times one third and then we have to Cuba's eight minus seeking cubed If I defy the data and now using food Beanies, the're, um this is equal to the integral from 0 to 2 pi di fada times the integral from zero to pi over three of here we have eight sign of fi minus and then sign of fi times seeking Cube Defy where we have signed five or cosign fly is tangent fi times seek and square to thigh Defy all of this times one third as well You're pulling out and so we get one third time's two pi times taking the anti to review here. Well, we have negative eight co sign of fi and then the anti derivative of tangent. If I seeking squared. If I well recall that the derivative of Seeking Defy is seeking fi tangent fi. So the derivative of seeking Squared five is to seeking five times See confide tangent five or two tangent. Five. Seeking squared. If I So I want to divide or multiply by one half. Seek and squared of five. This will give us Qianjin fires you in squared five when we differentiate and we're evaluating from Phi equals zero to pi over three. And so we get two pi over three times now plugging in negative eight times the co sign of pi over three. So this is negative. Four minus and then one half times the seeking to pirate three squared with the seeking to pirate three is going to be too second squared before is this is minus two minus and then plugging in co sign of zero is once This is minus negative eight minus one half times the second of zero well squared. Seeking of zero is the same as one. So seeking squared zero is one and we have minus one half. And so we get to pi over three times. So we have a negative four. Minus two is negative. Six plus eight is positive too, plus one half. So that's five halves which simplifies to five pi over three. And this is our answer. In fact, if you compare this answer to the answer from the previous exercise, it's going to be the same because we're calculating the volume of the same region and we see here that we actually use a different method to calculate the volume and you can decide on your own which method was easier. I think personally, there isn't a whole lot of difference there about the same difficulty.

Just before you with the region, uh, region. But he said to have these para veloute c is equal to five minus x 906 Plus why squared family was good. Minus these uh, the, um got her name between these Won't see she won't before for exist square Plus why squared? So these air to problem, it's on. Duh. When you look here word a three dimensional space Why six i z these this probably lawyer should look something like that. Sort of like these from the these one is gonna be So what are you here is gonna be five So something when I hide five on being well known so even obtain like a region like that I should be symmetric with respect to rotations Our Vera So So how are two bounds for these? You will be between those universal bubble bad more Knicks. It's why squared I really there is a happy above all these are below off these one life mine six were minus y squared when we can use, uh, cylindrical corner of the sun You see metre logistical mix Why c max investment with those soon. So so you can describe it through cylindrical coordinates because X squared plus y squared Elise R squared. So we have something in peace. So see how sweeting or art square y minus r squared so that allow those are the bones for C onto few year old. After which value goes, Are we help here? Speaker drawing over there so far I mean respect Z So you love these curve for our square is going to see with the Kirby's five. My sir squares of a d. R. Will go from zero up to this point. It's called These one are not so for these point should have that those two equations are equal. So we have four are square sequel to five minus are square r equals who are not so that we removed that they're gonna get miners because supply supply. I didn't that $5 Question five. So we get these desirous positive. The only solution is Article one are positive. So this point will be one. So I would go from 0 to 1. Uh, well goes over their own. You can rotate so the region can be seen us like you have these. You did it well. There the whole term So that said I would go from syrup to to buy a holder. So the volume the bullet was gonna be Why so for Syrian those bound So for within for our square arm five star squared movie See? No, no, the volume on this are are dizzy Would be that then for our she will from zero off the one on Thera from sear off to Dubai. Really? The seems, uh useful you before things independent of better. We can't theory within a separately. So they drove from 0 to 2 pi incident. That would be just Syria. I really didn't think to buy zero todo by. We're actually really that internal in these illegal to to apply times from zero the one from four Art Square five minus are square Are this you are huh? So all we have here here's being geo off this year The drill of the seat There is this constant with respect to see we just see so that evaluated meeting by my start squared before our square This is a beautiful tune. No, by minus R squared my hose that minus four. Our squares of these needs to be said it's up to my minus five to these. Or is that so? That these were turning to brown you by it was the interval from zero up to warn Fi minus five part square. We are So we're here. We have being drills. Well, the trouble So carrying to buy the journal five is just five are on the drum. Our square is our Q thirds by the bar rules another minus five nursing people to these evaluated one since you which is not me fool to buy them some five things one minus five times one cube. First we got minus Sear them by I'm seeing a cube thirds thes part zero His ear So with people too is boiling with people to find James five minus five hertz we can write that began. In fact, with five two times fly by one minus 1/3. Eso won with this one thirties will do so long as you will go any thirds. So is it even true? 3/3 miners, 1/3. The lightness like this one bring us warm. Third, to be able to sleep minus 1/3. These things, if I can find out there, is that So this is just too. Then they're still by That's these will be ableto to anybody. Good, like three. So we're this This will legal dio to the volume of these region. Well, about begin over there. So William, do any by over three.


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