5

Find fIL IdV, where R is the solid in the fist octant which lies inside the cone 2 = Vz? + y? and the sphere z? + y? + 22 = 1....

Question

Find fIL IdV, where R is the solid in the fist octant which lies inside the cone 2 = Vz? + y? and the sphere z? + y? + 22 = 1.

Find fIL IdV, where R is the solid in the fist octant which lies inside the cone 2 = Vz? + y? and the sphere z? + y? + 22 = 1.



Answers

Use spherical coordinates.

Find the volume of the solid that lies within the sphere $ x^2 + y^2 + z^2 = 4 $, above the $ xy $-plane, and below the cone $ z = \sqrt{x^2 + y^2} $.

I think this question be equal Triple integration or through square spine Boy, do you know Do you boy, do you see that on the integration limits? 03 Or you work for boy over to zero boy. By taking another configuration, we'll have mine sign if I defy the seat which is equals mind Ruto over to immigration off the Sita which is equal Mind group toe boy.

Following gone, You know, three dimensional space a purity eggs. The UAE on the C axis on. We have this cone. Very nice cone. Describe. Very question not, um see, Is he gonna do the square root off X squared plus y squared. So you cylindrical co ordinates. Does the Nazi sickle to our Onda want to compete? What is the volume off this cone between? Oh, making a cut through the plane c equals one. The plane she equals to two See equals two to see if you have Ah, a region like these. Like in beating those two plays. So these planes, these lower plane is described by C equals one they only see is able to do, um, she would have ah, this condition that Ah, Well, uh, see a sequel to our But also see these vine between 12 So, uh, we could make are are to the band. Um see, because, well, you're gonna look at at, uh, transgressive got, like, upper file. You have? These were We're see is here. Okay. I didn't want to and then are you go from zero bill. See? So Argos, uh ah. From zero off Seabees were bean. He are. I'm Dana. See, with the variety between one on too. So this is for the volume on. They know our angle goes all the way around. Yeah, because, uh, to obtain this fear these you will take these around data Your home All term from zero. The Dubai. Um, So you want to do with these? Will they have control of RDR? You took a large yard that is equal to our square house for the vehicle to, uh, during that c squared house. Um, I know cereal Europe, that on the individual off the sea we can one on too. There are three things here onto by Well, so, uh, be in trouble. C squared, you know, You see, drill the sequel toe CQ thirds so that Israel follow its beginning two on wine. We're gonna get to que minus one cube. Third on this is equal to so thank you. Got these, uh, age time stood. I'm still made my house one. So that is going to seven third's. Oh, so these would get us on our factor of one house from dinner. These will be so 3rd 100 times 7/3 feta. Where you can pull that out. So we have seven third's. Ah, you know, I'd like to Many dropped from 0 to 2 by the era on they drove. He said I just, uh, just gonna be better. I know that by zero all times that and so this will be able to two pi so seven times to buy. Ah, divided by three times to. So that would be equal to seven by thirds. Um so if he's up now, seven by third's, sir, my thoughts that easy, but is the volume before you want this region?

The problem is you cylindrical coordinates find the volume of the solid that is enclosed about corn. They is equal to the square, a tive x squared. Plus what's work on this fear? X squared plus y squared plus C squared is they go to to So, first of all, myself, there's two questions we have X squared plus y squared it secret who? One on behalf of artists between zero and one. Then the volume is because too integral from zero to high into girl from zero to one integral from far too squatted to two minus a squire R C e r sita, which is a two euro from zero to high interior from zero to one. Our time squared to to us why your mind square you are, which is equal to two high Holmes Negative third two minus r squared to the power of three. Over too, from zero to one on my watch of the author is oh pi Times Square image of two month, one over three

In this question we're going to find using spherical coordinates, the volume underneath. A specified array of shapes were given that we have to find the volume up inside a sphere above the xy plane and below a cone. So because of that, we need to figure out how we can set this integral up first. Now notice, let's look here at this equation here. Remember that X squared plus Y squared plus Z squared is equal to rho squared. Hence We can establish that row equals two. So I know that we have a ball that is centered at the origin with a radius of two. Now, remember from the book, we know that the cone. This is from the this is from the book. This is from the book. Okay. And what we know is that we have, we have the cone above the xy plane And this is just Fyi equals pi over four. Yes. It's a really useful trick to know if that with that. So you can draw that If you draw off three. Oh yeah, we know that we have we know that since we are above, Since we are above the xy plane, We know that the maximum limit of five has to be pi over two. So we know that Pie of reform is bounded And pie over to our our bounds for five. Yeah. With that being said, we can also assume that we have no. Since we have no data restrictions. We also will assume that Data is in between zero and 2 pi. With that being said, we can now make are integral are integral is over the region E. And it's just a differential volume which is we know that E is the set of all of all spherical points such that zero is in between rows in between zero and two. Thetas in between zero and two pi and Fyi is in between pi over four and pi over two. Hence our integral will become the differential volume becomes rho squared times Sine of phi dear! Oh, defeated defy so we have an integral from pi over four to pi over two. An integral From Syria to two pi. An integral from 0 to 2. You have real squared times the sine of phi times dear. Oh, decided to defy. We can separate these integral by multiplication. So we have the integral from 0 to 2 of Rose square. Dear. Oh, multiplied by the integral from 0 to 2 pi of one D. Theta multiplied by the integral from pi over four to pi over two of the sine of phi defy This results and we have 1/3 times rho cubed, going from 0 to 2 times two pi times negative. The coastline of fi going from negative from pi over four to pi over two and doing the evaluations, we get that we have this as We'll have 8/3. All right, 8/3 times two pi times. Yeah, co sign of fiber to zero minus coastline of five or four is square root of 2/2, which will simplify very neatly to eight pi route to divided by three. Yeah. And that's the answer to this question.


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