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3 Which of the following integrals gives_the volume of belowithe cone Iz 12| Vrz V2 labove] thel region plane and inside the cylinder (1-1)?+4 H the TyIdoo () K (2...

Question

3 Which of the following integrals gives_the volume of belowithe cone Iz 12| Vrz V2 labove] thel region plane and inside the cylinder (1-1)?+4 H the TyIdoo () K (2 _ r)r dr de I52 #Haa (b 6 6 M")erdr del 12 #l0 (c) 6 J Hade 12 hm (d) 6 % #hele| Ie fmno () [ (2_r) "dr co] M

3 Which of the following integrals gives_the volume of belowithe cone Iz 12| Vrz V2 labove] thel region plane and inside the cylinder (1-1)?+4 H the Ty Idoo () K (2 _ r)r dr de I52 #Haa (b 6 6 M")erdr del 12 #l0 (c) 6 J Hade 12 hm (d) 6 % #hele| Ie fmno () [ (2_r) "dr co] M



Answers

Let $D$ be the region bounded below by the cone $z=\sqrt{x^{2}+y^{2}}$
and above by the plane $z=1 .$ Set up the triple integrals in spher-
ical coordinates that give the volume of $D$ using the following or-
ders of integration.
a. $d \rho d \phi d \theta \quad$ b. $d \phi d \rho d \theta$

I have problem number 80 said the best way to do this is using Cartesian coordinates and then use a change of variable, so we're gonna set it up. Since the volume of the lips oId is eight times of all your the och tens with multiplied by eight. Integrate from 0 to 1 Integrate from zero to one minus x squared over a squared Have the integrate from zero one mice X squared over a squared minus y squared over b squared of one d z d y the x and then using X is you go to a U Why is it gonna be V and Z? Is he going to C W We got the new integral of eight times integral from 0 to 1. You grow from zero square toe, one minus you squared. Zero squared. One months, you squared minus v squared ABC D W D v d u. And since we can pull out, ABC is Constance. That means it's just a B C times the volume of a sphere of radius one, which is 4/3 pie. ABC

Well, I problem number 79. So using similar triangles. If we extend the trust them to a cone, then the height is gonna be Rh over R minus are And therefore the volume of the festoons gonna be integral from sort H and the girl from zero ar minus Z times are minus are for H Then we integrate from 0 to 2 pi of a d thera t a d c then plugging in our function Medicals pi over three times are squared class R r plus R square tons each.

Um, no. One thing I could think it was in the office, not in the office. And this thing, they have a car. Five different accounts. It's kind of a stop that car. Okay, okay. I just woke e They say everything twice by okay. E A Yeah, Yeah. Z a, uh, let me write down a car. E. I'll be back here in just a, sir. Is our internet off? Yeah, not yet. Can you Can you go somewhere real quick? Okay. Okay. Good for me. Yeah. Yeah. Okay.

So now you're following region. Cylindrical coordinates. Selene Ricco Corn. It's, um, virgin. He's, uh, nourishing between the plane. They It's of all these, uh, kun. They described the o see people, too. Square it off X square, plus y squared We are Or that cone busy. Then it's bounded by mound. Why, um, Sequels to my no sex school bus. Why, sport? That is like a parabolic region. You're he's there. So you want to see and to compute? What is this volume role in the fire? Also, we just want to set off the intervals. Um, so, uh, you want to compute being traveling disorder D. C he r D Sarah. Also, you have enough Proctor a buyer today. Jack Cogan, when you know, for computer nobody in Beijing. The Zeppelin is cylindrical coordinates. So our c who knows between Ah, you know me You drove, um Well, the what would be their radius? You respect the G. So we're gonna have this line here. That line represents, uh, we'll see equals square it off X square, plus y squared buddies are So are seeing is gonna be between that And then Now these huh? Well, you can below Looking looking. Ah, don't work. So, uh, this one is too minus r square toe. See? So that or z move, arrive untying between these number on dot Remember? So the band, uh, on our rich are we're looking at So that is gonna move between our tool minus r squared and so Well, um, r r goes from zero up to that point there. Arm. So all this point appear is when When those two oracle and see coincides with portals through. So you see, we all are has to be able to do minus r squared. So you move the arty side. Now these most of these manufacturers toe knows, uh to one minus R no. So about, uh, yielded like those two, you get minus to work, but started. Some miners are so well, these are goes off the one because they see her there on the world are feta goes a long way around from zero to buy. Um, see, I'm that said I was from here before I It's your goodbye now. You want to sit up being developed in the order, You see the RTC? Uh, he said I will be of the are going from the Napoleon being was You see this region? These, uh, this graph, um, we hear you have the Lina. Oh, I see. Equals R. Well, being below this point, which is gonna be the same point of intersection Siegel's one. No. The line to minus r squared so that well, for for signal, you think zero on one they are goes just up to see so that they should be doing zero z. Well, that is the bonds for or C excuse with zero one involved. There goes all the way around. You too. Bye. I'm the unit. Well, there is more because there's also the condition for C between one and well, the point which there is Sequels. Those two for one on two. Um, this video are are the sea. Um, these two are I want to see butter to Caesar That, uh, we are can be seen us. You have to my no see moving this year there. The order is goingto r squared so that our since Ari's always positive would be really able to do that. So are our goes from zero off the square root through my no see squared being Ah, there goes all the way from you by So that that is, uh you want disorder? You You need to consider those two internals not being in the last order. You have deep Well, the sea you are. So we have to include that on our Is that it's a fact of company from going in the servos from here to buy on da while we have the same order I was in here bc then you don't know. So she goes, Oh, uh, from, um she was from huh? She goes from are up to to minus R squared. So this is not zero with our for C goes room are up to two minus r squared and then our our booze from zero off the one you know. Yeah, but is it going for our Yeah, basically, these one is, uh, but be very similar to love you. Just that the order of he's in jail suits, But ah sees, uh, there's also again here. There is symmetry with respectable. Uh, we can interchange. They changed the order of interregional Vera because there were some angel. The rest of the volume doesn't depend on there. This region is symmetric Respect. What? The issues are on there


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