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Let S be the region bounded by 01 = 2, 02 = 4,01 = T, 02 3 2 , P1 = {,and 42 T in spherical coordinates What is the volume of S?...

Question

Let S be the region bounded by 01 = 2, 02 = 4,01 = T, 02 3 2 , P1 = {,and 42 T in spherical coordinates What is the volume of S?

Let S be the region bounded by 01 = 2, 02 = 4,01 = T, 02 3 2 , P1 = {,and 42 T in spherical coordinates What is the volume of S?



Answers

Volume Find the volume of the solid obtained by revolving the region bounded by the curve $y=\frac{x}{\sqrt{4-x^{2}}}$ on [0,1] about the $y$ -axis.

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.

I want to go see their region inside. Well, this year, given by X squared plus y squared plus C square. This is watering a do that. He says fear. Very nice. Um, instead of this but out of Oh, the ceiling there from he's one. So well, these see? You know, here is this fear were used through because two is great. The radius squared off. Do because two is square root of two squared Onda. Uh, also we heard before situation we have are very nice is here. Yeah. Okay. Why? You see, so have a The blaming of the sphere of reduce square is a tool. So you have these rice is fear that point. There is a square to to So you have that issue on being We're gonna removed the feeling there. Uh, brady is one out of it. Ah, so there were going obtain some region, so the region would be all these inside of that. They re in deliver decent cylindrical. According it's, um So this condition is ah better. Our square loves he The square. This is more now, too. On these, that is, uh, our squares. More than one. So So that we could do. Well, we could split this region, right? It is symmetric with respect, respect, Drill. Um, flipping it so we could consider two things. Well, the region that is that those for Well, see, Go. Uh, we moved out there squares more than two square so that the restrict too see positive you have these bound So he's gonna be two times the region. Give him by sea between Syria and squared off to minors are squared from day. Mom, um well, are would be us three. Um Well, I will be meeting one on this point there is going to so that you have those conditions. I'm being all thera Cumbie in young girls off their ice, free to rotate all the way around. So there would be anything zero earned to buy back. So with those, uh, conditions, the volume will be able to do times. Uh, well, uh, seeing goes from zero to where the two minus r squared. So you have are Z, and then our goes from one of two. It's a little too there. Uh, yeah. Goes from zero to Dubai. They're very nicely of this region. Um, you make the great. You see a lot. The rest is a constant with respect. Busy. So some things e following here, too. Miners are squared. 00 would be equal to these tcisa for the two minus R squared minus you. So sort of these internal would turn into intervals to minus are squared them's are he are you know, between one's good off to Are there between Syria by said to them that, um So here you can forget integral, We can do a new substitution. So well, these internal Oh, are are they will it these on the inside of people toe You do V with two minus r squared, then you would be no minor. Still are the are you said that, uh, these that he's over here. Rdr he's gonna be go to minors in the U house in this interim off, people to, um, the one off the square. You you pumps with another minus. So for the bounds here would be well, or is it about that war is the bar with use of these will translate into what issue either squared up to cheese. Um, should be to minus two. A minus squared off two squared, but he's zero. See you on Dana at one. Would be to Tu minus one square that he saw. What? And so these scenes are people too minor This interval. One up, zero. This cooled off. You you Are we going to sleep? And in the year b minus, Should one with the wheel house. So these internal I know that they're they're scared of you, is you drink three, house them stool thirds the, uh, 1/2 there so that his goals way below the beauty zero on one. So these, uh, vehicle to, uh, where would be the third? Well, I want this three hubs. My zero to be just 1/3. So a decent drill. Oh, he's gonna be equal to all of that. 1/3 of people, too. Serves the girl from zero to buy. You hear eso Well, um, the drill off the thing is, just later what we did, we came to buy zero. You love deserves there. So that on Dana, all these is equal to two by my no zero. So we'll be going to buy times tow thirds. So be served this volume for by third's. So that is, um, the volume. So for by Kurtz

Okay. What we want to step through today right now is to find the volume. So we want to determine the volume of a cylinder that has a, um, base. Um, that is R squared equal to two co signed tooth data. So if they base with if a cylinder with this base, which is ah Lim escape and then the top IHS the, um Z equal to the square root of two minus r squared. So it's actually going to be a sphere. Okay, um and so let's kind of draw this out to the best of our ability to kind of see what we're looking for. And so, uh, if we kind of quickly drawl a sketch to the best of our ability, um, so that Lynn Escape, um, is going to have this is gonna be negative square root of two. And this is a positive square root of two. And there's my Lynn Escape and the, um, this fear is gonna be something like that. So there it could be my base with that sphere shape. So we really want to focus on, um, we really want to focus all that, um, sphere and so or not this fear, but that lived escape that base to find our limits. So we know that on the volume is going to be basically for that volume is, um, the area of the base times to hide, right? And so is going to be the integral, the double integral, um, of actually that right, so this is gonna be to minus are square times are tr teeth data. Now here's the deal. Now is now the limits open to represent that that, um area Or that, um, the limits will be dictated d r and D fada with respect to that area. So now we kind of kind of look at that base. So that base is, uh, this limb escape, okay. And so that limbed escape is going to go from. So if we're gonna be integrating the our value is actually going to go from zero to part of that limb escape, and we're actually gonna do four quadrants. So we have this piece, this piece, this piece in this piece. There were actually two be easier to do for those so wouldn't go out from zero. With respect to our to just half of that, lim escape. So it was gonna be this square root Ah, to co sign toothache because we're all gonna be doing half of it. And then our fado will be going from, um, angle of zero to pi over four. I hope that makes it, um and so that's how I got those limits. And of course, room do be doing four of those because this would be, um, part of that quadrant from zero Teoh. Half of that women escape and then zero to pi before to get that quarter and then multiply that by four. Okay, so now let's see how we can, um, get this started. And so let me go ahead. I'm gonna go ahead and racist. I here, give me a little bit of room, um, is now to be able to integrate and will probably will have to do a u substitution. So I'm gonna go ahead and let you equal tu minus r squared and do you is equal to negative to our d r. Okay, so I have that already are. And so now what? I'm gonna go ahead and do so negative 1/2 do you is equal to or tr so may go ahead and do that. You substitution. And so when I do that and I do that Sorry about that. Um, we get the volume is equal to negative to the integral from zero two pi over four, the integral an Amish. I'm gonna go ahead and change my lower and upper limits here. So if, um, are ISS equal 20 then you is equal to two. So maduro to here and then you are is equal to the square root of two co signed to data. Then use is equal to tu minus two co sign toothache. So the upper limit becomes to minus two coasts. I of teeth, data. And there we have, um, you to the 1/2 you data. Okay, that might just kind of make it a little bit easier for us to do. Um, And so now what we're gonna do is basically, um, integrate. So we have this equal to negative 4/3 the integral from zero two pi over four. Uh, you to the three halves evaluated at two. And then to minus to co sign tooth data de data. So it's gonna count to get a little bit, Um cumbersome here with our integration. And so now we have, um this is gonna be equal to negative for thers integral from zero to pi over for of we, uh, to minus two co sign to fade it to the three, have minus two to the three has and in integrating with respect to data. Okay, um let's see what we can do here. How they need to do, um, some kind of substitution. Um, I'm gonna go ahead and look at Let's see if I can do. Um if I remember my trig identities, I have co sign square you is equal to one plus Cose I cheer you over to And so what I can do, IHS Let's see, um if I do too so I have my coastline to you. So you co sign squared you minus one is equal to Ko Sai to you. So if I could substitute this in, then in here, then I could probably integrate, so But first fall, that's kind of focused on Let me go ahead. And as we work on that, let's go ahead and look at this part. And so that is going to be, um A to to the three house times a four divided by three fada and evaluated from zero to pile before. And then we're gonna have a minus 4/3 then this integral of this mess right here. And so if we go ahead and substitute this in for co sign to Fada, it is going to get to reduce down to a four minus a for co sign. Squared data to the three has teeth data. Okay, um and all idea was wherever there was a coast on tooth data, I substituted in this to co sign squared. They don't minus one and then rearranged it. Okay? And so now what we can do is we're going to you, um, work on integrating this crazy thing right here so I can factor out a four factor 84. Um, and let's go ahead and reduce this down. This is gonna be eight times this quarter to over three. Um, and then this will be they, uh, evaluated at pi over four and zero minus. And then what we're gonna do here is I'm actually going to factor out a and four and of course, at four is raised to a three halves that is raised her three halves. Um and so this will change to a 32/3 by factoring at the four and into a three house of it and multiply it to this four right out on the outside. Um, integral from zero to pile before and then when I have here is a one minus a co sine squared fada time they one minus co sine squared beta to the 1/2 data Because I have something I have. I'm gonna have a one minus coastline. Score three edited three has power, which now is X to the first times and eggs to the 1/2 that makes sense. And then, of course, this is a sine squared, which will give me a sign Data. Okay, um and so now what I can do is let me not skip steps. So this is gonna be equal to a, um, to square with two pie over three. Want I evaluated at pie before minus this 32/3 times the integral from 0 to 2 pi over four. Um, this is going to be a sign data, So this is gonna be a one minus coastline square data times a sign data de fainter. Then when I'm going to go ahead and do is do a you sub. So, um, I do you said and I'm gonna let you equal co sign data. So d you is going to be equal to a negative signed data defense. And so I used that in here to be able to in a great this crazy thing right here. Okay. And so when I do, I get to time Disk rotor two times pi over three minus 32/3 times a negative co sign data because I distributed this being too here. Then the integral of sine of data is co sign data, and then we have a plus. So then I did a use up for this section right here. Until I have a use squared. Come with now becomes a you cube to this will be a plus, a ko sai Huge data over three and them evaluate friend zero pie before I hope that makes sense. Okay. And so now we're just kind of go ahead and evaluate and simplify. So this is going to be equal. Teoh two times of Scrotie to over three times pi. And then when I evaluate all of this at pie before and zero, we get a plus 16 route to over free, minus a 32/3, minus an eight route to over nine plus 32 overnight. Put everything over a common denominator, which is going to be a nine. We get six route to times pi, plus a 40 fruit to minus a 60 looks. That is terrible minus A. She'll didn't do that very well, minus a 64 all over nine. And there we have it. There's our volume.

Girls given to us Old boy equal one minus X squared power over Negative, huh? And war equal four We had what boy? Equal one minus X squared bone negative one and Roy equal four intersects ed plus or minus square root 15/16. Therefore X raise from minus square root 15 over 16 to x equal square root. 15 over 16 Since our girl war equal one minus X squared or negative half which is containers and also the care. If off X is bigger than or or equals zero in the given region, so will you quit integration e be boy every fix square the X Equal boy Integration square root 15/16 minus square root 15 over 16 Poor squared, minus one over square root, one minus X squared over two d x Equally boy integration square root 15 over 16 minus square Root your hangover, Dean thing minus one over one minus X square The X Equal boy Phoenix minus 1/2. Lynn one plus X over one minus x, but would square picks. It's equally it. Boy square root 15 minus. Oh boy, Lynn four plus Square Root 15. This will be the final inst


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