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Set "p, but DO NOT evaluate integrals in sphiericalleooralinates |forthe volumne of the given solids: The solid between the sphere p cOs @ and the hetisphere ...

Question

Set "p, but DO NOT evaluate integrals in sphiericalleooralinates |forthe volumne of the given solids: The solid between the sphere p cOs @ and the hetisphere 0 = 2 with 2 2 0.cor(b) The solid bounded below by the sphere p 2 c0s 0 and above by the cone2c0s $The solid bounded below by the =y-plane on the sides by the sphere p = 2 and above by the cone 0 = #/3-

Set "p, but DO NOT evaluate integrals in sphiericalleooralinates |forthe volumne of the given solids: The solid between the sphere p cOs @ and the hetisphere 0 = 2 with 2 2 0. cor (b) The solid bounded below by the sphere p 2 c0s 0 and above by the cone 2c0s $ The solid bounded below by the =y-plane on the sides by the sphere p = 2 and above by the cone 0 = #/3-



Answers

(a) find the spherical coordinate limits for the integral that calculates the volume of the given solid and then (b) evaluate the integral. The solid bounded below by the sphere $\rho=2 \cos \phi$ and above by the cone $z=\sqrt{x^{2}+y^{2}}$

So we are. It's very goal coordinates. Only you would like to find the body. All the following reasons I have here. Thanks. Playing here we have. Ah, this fear. So this is here is described by is given by the questions over here we have here Z x. Why? Um this is fear is given by roll equals to two. No sign off you, um, so that. Well, Joe, they're always in here, and they know it is founded. Uh, between these. See, uh, he's gone. So the volume is the Is the region inside those two. They love this. Come here. The green cone issue will do you describe what you see? She called to the square X square? Yes. Why? Square. Mmm. Beluga. You make, uh, Alexey projection off these fear what we have his, uh, was like, this is the corn that are on the x Z flame will be just our ex eagle to see. We say that you see that sex? The Yeah, he's What is it then, Mrs Fear? Rowe people school shine. We'll see. We're still in the golden minutes. Um, you say that you have some point here? Yeah, these so you have? No. Is this as his row example here is free this your axes on a new project? Daniel, wait. I'll be exact Ceases making a position these under excellently. So while these fear is like having this like these blue area mom Dina rotating that right day our own, uh uh oh. From zero to buy trees like these and then rotated. So all these vision there is described through Well, row goes from zero to up to co sign Hey, for so our role in this region is restricted between zero on the oh, Sophie on. Duh. As you can see, the Sango field doesn't get any. Well, it's between the sangu. Um, that angle. So that angle day is, uh, his angle is free. Such that are Well, yeah, Let's have These is satisfied. So you have grow sign? No gold signed. Awfully breezy, pronto. These business here will be called the road sign so that you can So you see that one physical duel I can't or feel. So have the ski people to, um well, bye over four by other floor. So that, uh Well, um, this morning with the volume of ah, he's off these reason These volume, what will be will be calculated tribute Drill. It's here by him for so no, our angle if you you see this goes around? Uh, yes. I said out from by. Over for two. This angle there is gonna go despite house so that you have you think, uh, city she's seen by over four Rhonda by he hops. I'm the robe goes because you Oh, selfie, it's Cyril Uncle, shall we? On the volume element in struggle, Gordon, it's his road sign. Well, fie, is it volume element or Jacqueline, you are You want to put? So, uh, when doing the reason to go? Oh, we have a minority rope. So we were integrated first First, um, the trial of Rose Queer you roll that is able to Roque you thirds so that it followed. Everything goes. I know. Oh, hey, zero will be We'll be going to just close cube thirds my issue. You there. So Daddy's gone on being the CEO. He's equal to the internal from 0 to 2 ply. Yeah, by force. Well, to buy house wolf. Um who sign 1/3 off beside a fee Q. I'm signed. Hey, you see? So all these into got forties into girl. We can do our u substitution. You'll recall your ego's cool Sanofi, then do you vehicle toe arms minus through fear. Shut up this interrupt Just this piece for people to miner's angel Ova. You cube, your mind is comes out of the so your cube you I'm diving. Juggle these, uh, you to the four hour You ready for? So, um So you have ah, you to be four anybody before or mine? Is that No. Yes, there is a minor there, there. And so are our usable to go sign a fee Movies by house affords. So we have far these tamale ingredient. Sign off my hopes are co sign by force. Um, so call Sign up my house, huh? Thes will be able to minus. Perhaps he's able to zero. So it was you full of better before Milissa signed off by Ford's you can do. Ah, cerebral check rush. You have my faults is like having the single lords so you'll get worms one so that cho sang of 4/4 would be these range that it is such a letter that length is also able to dad so that you go That and by the way, be worth here, and we have the square. Plus, the other length is also a good way. The square places question with ones that you do is question one. Hey, see you go one of our in half. So he scurries about one household. You're going to want more square root. So, uh, in my forties ah, one of our squares off to to have what we're squared off to on the done to the fourth power you write for, so that these minus and minus Consul, that is gone. So, uh, sort of you have This is gonna be ableto one. Then what happened in that? Because no, Um, so you have a, uh these are the C is equal to one today. Her thio to throw 1/2 and then race to before was goodly for have stands. 1/4 four houses people, too, to sweets. One, these numbers will go one or two square. There's two schools 1 to 4 times, for she's won 16. And so, uh, yeah. So these would be all these new hero people too. Oh, they're 1 16 right to have a mom. Yeah. Uh, right. So, uh, so then, uh, these horn general people, too. Once there from Syria by 1 16 casino. He said So that is Angel. She doesn't mean to compute. I'm going now. Well, that dynasty going toe won over three times, uh, 16 terms to buy. And so we'll, uh, please these through councils. 16 to them say to these two councils. That too. So this will be ableto by over three emceed. Oh, she's, uh, 16 plus Stays off. Well, by where? 24. So this volume. Oh, Maria. Yeah. Ball. You ball. You, Monsieur. Why were you in for before you of these millions?

For the following volume. Um, he's, uh, or is a bounded between the corn. Um, by this commonly described by he eagles to my hurts between its very goldkorn needs. Very cool. It is. Above all, the next life plane explains this quite lately, saying that Zizek all did you on your side of the on this fear of really usedto So you would have something like that. Sort of like these region. No. So that is this fear of release to So what you say? I love that and also roll small them too. So you like to book you? What is the volume of these? These volume. So, uh, for feed you know that he's gonna meeting by thirds on the single. But I'm going there. Something equals to my house. So, uh, feel beating by thirds. Um, by hops on roll go from Syria to on, uh well, fair goes totally around. Were they not so the whole turn from zero off by two to play. So the volume is gonna be able to, um so fair goes from Jodi by buddies. The bombs 40. She goes from 5/3. I put my house. There is defeat on row goes from 0 to 2. You know, the Devil UMA Women Peace role Sign of feet. This is the volume element. So Marie melamine is very so you need a great, uh, it offers because nothing here depends on finish. He said, uh, from zero. Dubai is just gonna be equal to theta when we did to buy on a zero to would be able to just to buy So all of the central thing that movie go to to buy times drill from by thirds off by house. Oh, literal from 0 to 2 off road squared, sighing the row, the feet No. So, no, the reading throw room. You have no control. A rose clarity grow that would be able to roll cube thirds. So we are doing that. Taking too, and zero you don't pain to with their power hurts minus hero. So we'll be able to that. So we'll have that these people to to buy times Lewis there blower be very by three and then we help the last into Roland Girl from my third something by house Books Idol. He few room for the drill. A file sign draw sign where hes minus co sign Mine is close cause you view the friendship because with respectable See the thing you have fresh with minus call, Sign the pain sign. So, uh, you would be is regal toe these internally will do to buy them stew cube over three times, minus so close off by house, Myers cause by thirds I thirds. Uh, So who signed off my house? About zero. Dana. They have minus minus. So well, we just less well, that comes. We don't know what you're saying. Co signed off by thirds the sequel to together you circle your This is one, Then we'll by thirds. He's like these angle because, uh, he's, like, dividing how I turned out the spy into free free slices so that we're gonna have, like, one toe. So these families, 5/3 I'm doing well, it's cool signs. You will do 1/2. So this is one, uh, nice about, Of course. Why third's So these were the 1/2. So miners might as 1/2 of these 1/2 councils want to there. So you obtain that we should be gold miners. Why? No, that's positive. So good. You have ah positive number which we sure is a volume. So it's gonna be do. Their power comes by thirds. But did your thirties going toe eight eight by third's mom? So that should be It's about me life paid by thirds.

And this problem, we're going to find the volume inside the blue half sphere and outside the purple cone. So the sphere is ro equals two. So that tells us that row goes from zero to row equals two. Because you're looking at these little pieces like this fee. Okay. Sophie goes from the cone, which is pi over three to the X Y plane, which remember is pi over two. Because it's from here to here. So we have power three and then pirate, too. Zero equals pirate. Three to row equals pi over two fee. I mean, poverty, too. I got so confused, I made a messy fee. Their fee and then data data is we want these little pieces to go all the way around the Z access so theta will be going from zero to two pi. You notice this is symmetric around the X axis are around the Z axis. It's it's the same in all four top opt ins. So I'm just going to do four times zero pi over two. So long you will be four times. Yeah. Okay. One because we're finding volume with triple in a girls rose squared sign fee de ro ro dif e d theta. Okay, because that's the integral constants and things that you use. Okay, fee is Rose going 0 to 2. Fee is going pirate three two pi over two. And since we got a four, there were going to go zero pi over two. And okay, so then the first hopes for first dinner girl is Rho Square D row. So that'll be row cubed over three from 0 to 2. And then we still have the sign fee D v d theta. So row cubed over three from 0 to 2. That will be eight thirds. So now we have four times eight thirds in a cruel, integral sign fee open meant integrate their defeated Thera. So 32 3rd integral of sine fee that is minus the coastline of fee from pi Over 32 pi over two. Do you data? So that will be 32 3rd integral minus the co sign of pi over two minus minus. The cosine of pi over three d theta. Oops. So that's 32 3rd inter girl CoSine of power to a zero minus minus Makes positive. Cassina pirate 31 half data So Now we have 16 3rd integral zero to pi over two, Do you theta? So the integral of data is just data 16 3rd data zero pyre or two which will give you 16 3rd times pi over two, which is a thirds pi.

We're getting a great y squared Z squared over this um shape. So above the cone Row equals pi over 3? So it's a really fat cone. Um because power for three is 60°. So from here to there is 60° And below this fear row equals one. That's the green thing. Okay so row is starting here at zero and going up to the sphere. So row will be going from 0-1. Oops I meant to erase the mess I made there. Okay row equals zero to row equals one fee. Always starts here at the Z. Axis. So this is equal zero and then it's going to stop when it gets to fee equals pi over three because that's the answer to comb. So that's what's keeping it inside the cone this fee equals zero. Duffy Equals five or 3. And then tha tha well it's the whole cone. So data has to go all the way around to get the whole cone. So data 0 to 2 pi. Yeah. Okay so now I'm just gonna leave those hang in there. Let's see why squared. So why is rho sine fee sine theta. And that's squared. And then Z squared will be row co sign fee squared. And then devi is rho squared sign fee dear O. D. V. D. Theta. All right so here we have a row squared. A row squared in a row squared. So we have road to the 6th and we have a sign fee squared and a coastline fee squared. And another sign feast. So we have sign cube fee cosine squared fee. And then we have a sine squared data. Okay that's only one of those dear. O. D. V. D. Theme. Okay do you have the morning? Okay so we can go ahead and integrate row to the 6th dear. Oh and we'll get Road to the 7th over seven from 0 to one. Thank you go some squared sine squared. Do you feed the rope defeated data? Okay so that gives us 1/7. So let's put that out here. Yeah. All right now here I have a odd number of signs and an even number of co signs. So I'm going to take one of these signs and split it away from the other two. So I have sine squared fee, coastlines squared fee sign fee D. V. And then we still have the sine squared theta D. Theta. Yeah. Okay. The reason I did that because I want this to be D'you so I'm letting you be the co sign. And so d'you is minus the sign. So I need a minus sign in here which will put a minus sign out here and I need to replace the sine square. So now I have line is 17 integral integral 1- Cosine squared coastline squared minus sine squared minus sign. Sorry the fee. And this one was the was 0 to Pi over three. Okay if you equal zero I'm sorry if uh V. Equals zero then you equals the cosine of zero which is one If he equals pi over three Then you equals the cosine of pi over three. Uh huh. I have three square 321 Which goes 1/2. So this turns into 1- one half. One minus U. Squared U. Squared D. You. It's one half that's U. Squared minus U. To the fourth. D. You. Okay so you cubed over three minus you to the 5/5 from 1 to 1 half. So 1/8 over three -132/5. Mhm -1 3rd -1 5th. All right. I'm gonna work on that over here. I got one 24th minus one third minus 160th plus one fifth. So this needs an eight and an eight. So -7/24. And then this needs 32 minus plus 31/1 60. I can't Mhm. I'm just gonna do this minus seven times 1 60 plus 31 times 24 Over 24 times 1 60. And then I'm going to reduce instead of trying to sit here trying to think of what the common denominator is. 160 times seven so minus 11 20 plus 31 times 24 7 44. Why nestle. Okay, so now I have 376 Over 24 times 1 60. So I'm going to divide by 218, 8 Over 12 times 1 60 94. six times 1 60 47 over three times 1 60. I'm not liking that very much minus one. Seventh times 47/3 times 1 60. Hope that was negative. And now I have to integrate the sine squared of theta. D. Theta. So 47/7 times three times 1 60 I have to put in an identity one minus the co sign of two theta over to this data. So 47/7 times three times 1 60 times two Integral of one with respect to data is data-. Can we need a two in here to make the you and another one half. So one half signed your data We're going 0- two pi. So 47 times seven times three times 1 60 times two times two pi. Mhm. Yeah seven time oops seven times 3 times 1 60 16. 33 60. Dude. Yeah okay let's see if I can. Yeah yeah yeah okay so I paused the video I'm not sure what that looks like on your side. And I did I recalculated and I got minus 47 of 4 80 so I'm not sure where this seven came from. So we got 47 minus. Uh huh. Then somehow it turned oh yeah it's this 7:00 47/7 times for 80 times pi. That's what I get for the answer. It's kind of gross but I can't find anything else wrong


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