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Sketch the region bounded by the surfaces $ z = \sqrt{x^2 + y^2} $ and $ x^2 + y^2 = 1 $ for $ 1 \le z \le 2 $.

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Johns Hopkins University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Okay, first let's draw these things separately and then we'll put them together and make one picture here. First of all, Z equals the square root of X squared plus y squared. Look for a minute. Let y equal zero. Then you get Z equals the absolute value of x squared. Okay? So in the Xz plane, which is this one we get this line to Z equals x. Z equals X. When x is one, Z is one. So it looks like this except positive Z. Only because it says positive the square root. Okay, so it's 45° in the XZ plane. And then when you let um X equals zero, you get Z equals the square root of y squared, or Z equals Y. Which is also a 45° angle. Uh that's a terrible picture of me drag getting here. Yeah. All right, here's my coordinate system. So the first one is 45° into the first quadrant. Okay, right here and then the 2nd 1 45 degrees this way. So right here. So what this is a picture of is a cone that opens up on the positive Z axis because this is positive the square root and it's 45 degrees up from um the xy plane. So 45 degrees here, 45 degrees here. 45 degrees here. Alright, next x squared plus y squared equals one. Well, in two dimensions, that's a circle whose radius is 00 and I mean center 00 and its radius is one. So it's the circle right here. That's the circle like thing. Right here, one more time there it goes. Okay, put them like it's a circle. Okay? And it doesn't say anything about z, Z can be anything. Nazi is nothing Z is anything. So, what this is is a cylinder. Okay, So it looks like that and Z equals wanted. Is these between one and 2? What? Okay, z equals one. That's the xy plane, Z equals two. That's the plane parallel to the xy plane. But two units up. All right. So, we got a cylinder. We got uh cone and then we got this cut off front so that it's above the xy plane and then cut off at two. All right, So, let's see if we can get that. Okay, I think would be useful to draw the cylinder first. It only needs to go up and then inside the cylinder we have this cone And then we're gonna cut it off at um Z equals two. And when z equals two, let's see where we are on the cone. two equals x squared plus two equals the square root of x squared plus y squared or two for equals X squared plus y squared. Yeah. Okay, so let me think about this minute, let me okay, what I really want to do is start my picture over. Now, I've decided to draw the cone first so they can get a bit drawn better. Okay, It starts here at 00. When it gets to Z equals one, It's a circle of Radius one. When it gets to Z equals two. Now it's a circle of radius force, so it's way big out here. Okay? So there's the cone and then the cylinder it's cutting right through here. Okay? There it is. Okay, so in the picture We have the cone up to where it gets to one. Z equals one. And then after that we have the cylinder until it gets too too. Okay so it sort of looks like a pencil or pin like that Where this is equal zero, This is equals one and this is equals two. And it's cut off here because the cylinder is smaller than the cone is once you get above one. Okay.

Oklahoma State University

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