convert-the-integral-int_-11-int_0sqrt1-y-int_0xleftx2y2right-d-z-d-x-d-y-to-an-equivalent-integral-

Question

Convert the integral
$$
\int_{-1}^{1} \int_{0}^{\sqrt{1-y}} \int_{0}^{x}\left(x^{2}+y^{2}\right) d z d x d y
$$
to an equivalent integral in cylindrical coordinates and evaluate the result.

Carlos Pinilla · Accepted Answer

a full intro in Kardashian Kourtney&#x27;s syndrome Using dropped from my one on one and each other from you hear the one minus squared off screwed of one liners White squirt on from zero X off X squared plus y squared. Oh, I see. This is about for C X, the ball for excess thes on the y That is about for why so, um like to turn it into our doing cylindrical ordinance are computed. So, uh, we have everything. First, how&#x27;s that region look like? So we have Ah, something like these you have here z the X and the y axis. Seen that need Z goes because you&#x27;re next so late you that is there not so strict plane seeing people to eggs It would be between the stools, abounded like that plane on being for X. You see, Iran, uh, squared bullsh, uh, one minus y squared so that we see that, uh, be square taxes. But I mean extra scores that so it is these region, but, uh, So this part of this circle, but for why? Beating miners one on one. So just God part. So the like, these region, she would be doing internal these this vision. So you say that, uh, circle on the plane. So we will get that is cylindrical. We should have our angle. Said, uh, he&#x27;s gonna go from that angle up to that angle to be from more. It&#x27;s cylindrical. Fair is gonna go meeting, uh, minus my house on, uh, buy a house, Onda. Uh, well, our radius goes from Syria off warm, because these, uh, I was trying to be instead of that. So our goes from zero after one on, uh, this convention. The conditions that are she is between sear on the X with legal access. You need to go card called Invisible to our course. Yeah. So these intro will be we&#x27;ll turn you to the integral are more. See both within zero. I&#x27;m our course. Vera 36. Uh, these political coordinates was gonna be r squared on the these this part turning to our the sea we are because the volume element are and so, uh, are those from 0 to 1 from issue of Born on feta lose from rightness by house. My house minus by halves up by helps. So these is the internal. Very nice. Now, Um we go ahead on beauty. So, uh, well, firstly, interbreeding Z, but as a u z. So, uh, just evaluate in there with the reversible we have on our cube from there, being times are course, Vera. So you have that on being your integrating our You are the fair miss by halves up to my house. Oh, by house. So, uh, we do it here that we are the fourth. The trouble are to the fourth gr It&#x27;s gonna be rto five new anyway. Five. You can see if you want to. Probably be Dean. 10 The birth. She&#x27;s one minus you. So, um eso that is gonna give us, uh, one. We&#x27;re fight times 1/5. Course later. Sarah the king, minus by house. Earn my house. Um, so the drill, of course. Science fair. Uh, the central physical tool shine. So it would be ableto Oh, see through sile by halves. She&#x27;s having porn fever stems Sorry. By hubs. No minus no sign of Wales by house. From the look of our paragraph sign. So sign is like the something. Uh, my heart is going toe one, Simon. Thanks from the but minus Phi house with minors. One is mine. Is one there? No wonder. Said that, uh, is ableto one miners. Mine is one very like five should be going to to gives, um said that, uh oh, That should be already drill over five.

Kelly Hong · Answer

All right. So are doing number 2 67 And we have this inner cry here that we want to change into cylindrical coordinates. So the first thing that we wanted to sketch the region of integration Let&#x27;s do that over here. I actually want a sketch entity because, well, you&#x27;ll see it&#x27;ll make things easier to visualize. So if our exes go from negative squared off on my slice square to square it up on my A spice girl and I always go from 0 to 1. Then we have the circle of the here, and our region of integration would be this over here in a region of integration, birth disease will go. Disease would go from X squared plus y squared to this group of experts plus y scarred and in cylindrical coordinates. We know that our squared equals X squared plus y squared. So for Aziz, you go from our square to square root of our square, which is our So let&#x27;s rewrite this interval. I will also now we haven&#x27;t xsi in the vision in the function here and we know that X equals o r co sign data. So if we simply just put that in. We have our constant data to see. Now remember, we also toe add an hors D or data. So if we do that, we get our square. Costa did Rosie. That&#x27;s what that Deasy DRD did us right for About that the DZ. So we have the inside of the interval figured out now. So I have to do is figure out what the boundaries are. Super disease. We already said that disease we&#x27;re gonna go from our square toe are right. So the next thing we want to figure out ISS are ours. So if you look at the function our region of integration, her article from zero out to the function here, I already know that this is one all the way around because this this right here is the unit circle. So let&#x27;s go from 0 to 1 all the way around this circle. It&#x27;s always going to be one so or ours. Go from 0 to 1. No, If we figure out our theaters, we see that our circle goes from here. Over here. So this is the zero. And what are halfway through this wrinkled and this is pie. So your data is go from zero to pi. This will be your triple integral in cylindrical coordinates.

Carlos Pinilla · Answer

All right. So you want to compute the following to go from when you weren&#x27;t from zero disquieted of one minus y squared? Well, during the shear off X wolf. Exclude. Plus, why squared, huh? See, um, Yanks You. Why? So, uh, was wait to start is with all join and problems that religion look like to love here the sea axis, the ex lax season. Why? Access. So are saying a the rex. Well, ours eagles from zero Twix toe for the plane. Why? See that he&#x27;s, uh, line. So you&#x27;re just lying of in It was a na Siegel&#x27;s, uh, goes from zero goes from zero up there, so it will be see, like these, But ah, well, Rx was gonna go between zero on. Uh, why in my nose this good off? Why so well, for For what? It was you it would be. Think up to there. And so, as you can see here are why goes from miners? Want one? Well, so these region in the you tell here Well, your ex on why taxes are inverted us, always like, uh, he&#x27;s over part. Well, that should be, uh, circle. The supper parties described so that one minus lies. Where is you&#x27;ll go, x. So that, uh, will be I said I&#x27;m like these circular region, but, um, that is, uh, he&#x27;s got so that, uh well, your your c at every point. Uh, you know, buddy, what do you have with your ex? So what? We should be sort of like these, Uh, which So some really like that. And so Well, we I want to set it up in cylindrical. Um, well, these, uh, this Yeah, you I will turn out into our my will are you see, we are, uh, me here. Rhonda was You can see here they are goes from comes D&#x27;oh! Well, there would be. Hey, you should be like me. So we have Ah, here, Z one hour. Well, they are with me between you can be speeches, so he should be everything. See on, uh, is the line. Are you busy? So of Eugene on one. So is there He&#x27;s out Order to make it. Uh, you are dizzy. Um, So, uh, what could be use their See? It&#x27;s his, uh, one. So you have, uh, are are the sea so si goes from she&#x27;ll to one we have. Ah, is, uh, these regions on so well, they are his version from zero to Ramsey. C equals hard ideas that line 21 What? Um then the well, z goes from 0 to 1 when, uh, you can see here, Uh, you only have ah, have a turn. Oh, so these horrible you sound, but so, you know, move room sear off the play, and then we&#x27;re missing the function. That&#x27;s our function here. That is precisely all right. Square so that these interval will turn out into Well, we need to Otto when there is quicker. Um, shut up there for our drove with these, so we&#x27;ll be drilling is our Q. You are. So you&#x27;re doing that? No, I didn&#x27;t think you start to the fourth. Ah, divided by four. And so what we have to value It&#x27;s dad with Dean, See on one so that we would get full this first in zero R and see one of our u He earned her the sequel to one minus, uh, see there for for our very way for So we have Ah, You&#x27;d, uh, you turned the scene draw until angel from zero to buy off. Um, Angel from zero. Up to one. Warren minus C before, like for. Well, I see you get up to know. Well, you see, Diego could be donor thing to go, although it would be angel of 1/4. You see? Must the drill off minus four. Um, murdered my fort. So Well, so these one he&#x27;s just see And that b minus see there if our divided by four times five said that evaluated routine. Okay, See you on one. So no. 01 This is a warrant to go either spent you hear from Syria by sort of the This here is, uh, four. See you. Bye. So all these difference inside the sequel to 14 minus why, over four times five minus the difference of Do should be well, when a zero plus Cyril. So these pardons Xerox nothing. And so well, these difference here that is equal to all right. By minus one over for £5 force of these ones you will do he&#x27;s number appears for shouldn&#x27;t goes through castle. Then these internet will be You need to go off 1/5. Um Well, uh, the feta issue on by your regal toe. Why were what So these, uh, that he&#x27;s, uh, these intervals until All right, Well, what

Ahmed Kamel · Answer

this question. We have terrible integration off ive or negative room for three room over to signed Fight Digital de Foix be seated and the integration limits 01 Siegel by over two. Legal boy, we&#x27;re taking the first integration. We will have 1/3 one minus ive or negative one. Double invigoration off slang foi defy peace either which is equal boy over three one minus evil or anything.

Sarah Murray · Answer

So in this problem we want to evaluate the triple integral Given, which is negative one to run from the negative square root of one minus y two square root of one minus y And for negative 1 to 1 of X squared plus y squared to the three halves D c p x. Do you want first thing we want to do is convert to cylindrical coordinates. Remembering they are squared is equal to X squared plus y squared. You can see that side here. And if we take a look at our graph, we&#x27;re gonna notice that are coz from 0 to 1? Yes, data goes from 0 to 2 pi and it is he because the negative one 21 So if we make that conversion, this is gonna be r squared to the three half. If you are Are do you see evaluated from negative 1 to 1 p r evaluated from 0 to 1 in di Sita evaluated from 0 to 2 pi and now we can integrate inside out like normal So we want to integrate. Well, let&#x27;s simplify this First 0 to 2 pi you&#x27;re the one negative one toe one r squared to the three houses are cubed times. Ours are took the fourth T c g R teeth data. Now we can go ahead and integrate. Want to integrate with respect to Z first, So 0 to 2 pi 01 We are to the fourth time. See, from negative 1 to 1 TRT data. It&#x27;s gonna be pure 02 pi 01 and that is far to the floor minus negative. Or to the fourth t r T Seda, which is of course, going to be equal to to our 24 TRT data. Now we can integrate with respect to our co appear going to be equal in a girl from 0 to 2 pi and then, of course, to are to the fifth over five from 0 to 1 teeth data it&#x27;s going to be equal to 0 to 2 pi of to over five de seda and we can do our last integral with respect to theta is going to be equal to 2/5 data from 0 to 2 pi which is equal toe to over five times two pi minus zero equal to four pi over five

Kelly Hong · Answer

her surgery number 2 85 And we have a suitable that we want to turn into spherical coordinates. The first thing I wanted to sketch the region of integration. So if I actually go from 0 to 4 and are wise, go from zero to Route 66 square, then you can gradually this sort of quarter circle heater with a radius of for And if we have negative Route 60 minus X squared minus y squared to 60 mice. Route 60 My sex with my last word on the disease. You can imagine this disks Cruz sort of quarter of a sphere. So if you want to move out of two Sprinkle co ordinates. So you know that in circle coordinates extra for five scripless he&#x27;s cleared equals zero square in Red Square Square Biz wrote to the fourth power. But because we know that TV equals Rose Corson fee actually going to end up with Roe six Power sci fi do you row? You see data. Next, let&#x27;s figure out our boundaries for Data&#x27;s so we know that we&#x27;re calling from zero 1/4 of the way through so we know that this is Euro. I have next. Let&#x27;s figure out our fees. So you know that we have a sphere. So Okay, so this piece we&#x27;re not end up with zero all the way down to pie because there&#x27;s no limits on our disease. Finally, let&#x27;s figure out a rose. No, I don&#x27;t always go from zero hope. Words is going to hit for every time because it&#x27;s sort of 1/4. Is she atmosphere? So our rules go from it&#x27;s your to four in this hoping here with your triple in every in spherical coordinates.

Problem

Set up the iterated integral for evaluating $\iii…

Problem 36 Hard Difficulty

Convert the integral
$$
\int_{-1}^{1} \int_{0}^{\sqrt{1-y}} \int_{0}^{x}\left(x^{2}+y^{2}\right) d z d x d y
$$
to an equivalent integral in cylindrical coordinates and evaluate the result.

Answer

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