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Find the volume of the region that lies inside the sphere $x^{2}+y^{2}+z^{2}=2$ and outside the cylinder $x^{2}+y^{2}=1$

$\frac{4 \pi}{3}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Missouri State University

Campbell University

Baylor University

University of Michigan - Ann Arbor

Lectures

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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

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In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

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Sphere and cylinder Find t…

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Find the volume of the reg…

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Find the volume lying insi…

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

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