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Compute the volume of the region inside the sphere p = 2 and above the plane = 1_...

Question

Compute the volume of the region inside the sphere p = 2 and above the plane = 1_

Compute the volume of the region inside the sphere p = 2 and above the plane = 1_



Answers

Use spherical coordinates to find the volume of the region bounded below by the plane $z=1$ and above by the sphere $x^{2}+y^{2}+z^{2}=4$

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

In this problem. We are focusing on multi variable calculus specifically with integration. And we're told that we have to find the volume of a portion of a sphere. And we're told that row is less than or equal to two. And our plane Z equals one. So I made a very rough diagram to kind of demonstrate what we're investigating in this problem. So this would be our y axis. This is our Z axis. And imagine that this is the portion of our sphere where we see two on each side. Then rz access was this top dashed line. So Z equals one and then we see that this would be fi over three. So we can now, Owen, pardon me. This fear we can say that row equals seeking of five. So we can now bound Phi and Theta Phi is from zero to pi over three and theta is from 0 to 2 pi. So now we can set up the triple integral in this case of the volume, um, in order to find the volume of that portion of the sphere. So our volume is going to be equal to the Triple integral. The first integral from data of equal to 0 to 2 pi the second integral from five equal to zero to pi over three. The third Integral would be from row equal. Speaking of five to we take the triple integral of row squared sine of fi in dear oh defy d theta so we can evaluate this now because we've now set up the integral that we need to solve. So we would get the integral from Fada equal to 0 to 2 pi times the integral of Fada equal to zero of pi. Over three of now we would we have found the anti derivative of the first function. You would get row cubed over three squared a pardon me growth row cubed over three evaluated from seeking a fi to to pardon me and then we would have still left over sign if I defy Di Fada and then because simplify this further, we can pull out that one third from the anti derivative of our first function. We would get 1/3 times the double integral first inter girls from Fadi equal to 0 to 2 pi the second Phi equals zero to pi over three of eight sine phi i minus seeking cubed. If I sign Phi in defy d theta, we could simplify this again. You would get one third times the integral double integral the first one being from theta 0 to 2 pi fada equal to zero of pi over three of eight sine phi I minus the tangent of fi seek and square to fi in defy d theta. From this line to this line, we did a simple trig and metric substitution just simplifying the fact that we have seeking cubed in that case. And then we're going to let you equal the tangent of five. So that means it's derivative. Do you would be seeking squared of thigh in D theta. So we're going to have our volume now equal to one third time's the integral from Fada equal to 0 to 2 pi of negative eight times the cosine of fi minus tangent square to fi over to evaluated from zero to pi over three. With respect to theta, then we can simplify this by plugging in our A and B value. We would get one third time's the integral from Fada equal to 0 to 2 pi of negative. Four minus one half times three plus eight. Remember, this is with respect to Fada. And then we would get one third time's the integral of 0 to 2 pi of five or two in Di Fada. So we can simplify that To get that this is equal to 5/6 times theta evaluated from 0 to 2 pi. Then we're very close to being done. All we have to do is plug in zero and two pi for theta. And look at that. The volume of the portion of the sphere we were looking for is five pi over three. So I hope with this problem helped you understand first how we can Geo, um pardon me how we can graphically interpret the conditions that we were given where we started with a plane and a bound row value. And then we have to interpret number one what we're finding the volume of, and then put that into the language of multi variable calculus. Go through the triple integral in this case to find the volume. So I hope that this helps you understand that process and how we go through the triple integral

Oh, yes, we have. But these Aubert hemisphere worry the zone eggs. Who? I see access. You have. Ah, that, monsieur, I was really You. So do the others, too. On the way. Make your cut. Siegel's one for a plane. Well, please. That it was gonna give us. Ah, this is the plane. Um, c equals once we want to computer greasy volume of these vision in there. So it's you will be the reasoning inside of ah, this year. So it's here, So mmm. Roll. Listen to on both, uh, well, this plane Sequels one. Um, So how do we do that? So? Well, first time you have here this sea access and these two and one now here. Yeah, exactly. Said, I love you. Why you call zero. So now this is fear. Something like that. So there are love. Not which this happens. He's at, uh Well, um, you can't theory that they where these joint, huh? That should be like that half turned airplane ready by three. It was like, uh oh, this is so hold these angle. There is B equals by thirds. So that leaves is ah, this point, uh, also that, uh, scene should be zero on you. Bye, thirds. Um, you do the old tradition around Syria because it's a mature, respectful. So that was from Cyril off to buy when they know our radius role. So as you can see here, hero of the sea axis rose from one I talked to. But, uh, as you give one stretching there the range or roll. This is so rule. Uh, so they playing? See, those one is getting by or this is grow Call sign. Well, see, it was one you have to be or both. Well, that plane Ah, uh, she so rule has to be bigger than one American. Sign off key, Ana. So, uh, well, so you can check that on. Uh, well, you can check by. He's looking at zero one. Huh? Huh? Uh oh, yeah. We're always between zero on two. I've been at these points. Bye thirds. We're gonna have by thirds on the roads. You go nowhere. It is because one of the local sign off by thirds is equal to 1/1 half mexican too. So you would be saying with these equations that role has anything to want to Jesus. There is no space. So about that. Is it bound for rule on that? This is, uh, she can trophy. So these, uh, computing these volumes we can use very cool. Cordy. It's so that her room was routine. Sequined. Well, fie on two these zero natural squares. And he then, um, fee goes between zero and 2/3 cereal by thirds feet, and feta goes all the way around Syria to buy so that our leaving to be first row squared row. But he's, uh that is equal to broke. You thirds. Some of these will be able to well, that volunteer meeting to second. Do you see a totally with people? D'oh! 1/3. Oh, um two. Okay. Cube rule. Y nos See you again. Oh, you're cute. Monday. And, um yeah, that's it. So he if you feel 19 No, but, uh, you know, multiply by, uh, sign sign from the 18 0 dy hurts. Now it is your room to buy. Um, so we're really Dribble, dribble, sign. First internal There. That was a constant is gonna be equal to minors co sign. So that awaited Bye. Thirds and zero B minus co signed by thirds, minus sue. So, uh, all these number that he's done half miners want to these minus one, huh? We don't mind us a toe physical, too. I know one tough. Was it? Um, so that is, uh, these introduce of that. That would be one there Inter forms here by from dean bodies to the third. That's one house. Well, we have Lisi to lift. Um, Also these do feta. And then miners and drop bombs were took by on the trail from zero. No, Brian, this is here. Uh, if my thirds over sign off. Definitely big, um, seek and you will do one of our local science. That would be you are a vital sign. Q. And so here we can do a new substitution because, uh, because ever co sign off legal, do you, you legal toe mind sign. We'll see. Uh, so So a decent girl would be you for you squared? No, you're cute. Um, this is really the minus sign. Well, so the troubled minus Sino fee. This you quick the same off the cube. We'll be able to these with you, because to cool sign. So what, you're really doing these, uh, that is equal to Myers. Hey. Oh, our way. Um um You squared. So we have a negative Power is a minus one over to you. It's quote. Yeah, you differentiate. So she would be that evaluated, Nitin? Well, little sign. So it would be minus one over two. Assign a fee. Uh, squid. So that, uh, that body meeting by thirds you Bye. 3rd 0 will give us Myers Way. Doesn't have whatever because I know my third sees one, huh? Word. Well, then, minus, um wasn't a Jew is one on one, So these would be just two squared minus one. So these numbers these, uh three so minus you are to minors. Three house. Oh, so that controlled these minus three halves? The central here. These homes, We've been Mina's. He's minus three house. So that, uh, well, putting, putting everything together. Oh, what? So our little angel is one third intel from zero to buy? Oh, uh, so these term would be two squared. You know who's in the white of the squared? We're, uh ah. Well, these term waas was that my mystery house? My monthly halves. So? So they should be our volume from then. Uh Well, you drill off the hair from your device just to buy me to fly. Hurts involved. These numbers burned its study itself for runs the house with four people. Do you have more interesting house of V five bucks? So this is B times five homes, General. So is volume is five. Bye Herds. Oh, so really, all these reason highs by thirds, but

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.


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