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(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with $ n = 4 $ to estimate the volume of the solid.

(b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with $ n = 4 $.

a) 196 units $^{3}$

b) 838 units $^{3}$

Applications of Integration

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Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

suppose we sent Rotate this shape around the X axes, uh, find the volume of the region. And we're going to do that by using the midpoint rule with n equals four. So this goes from 2 to 10. So that means the width of each one needs to be to Okay, so here will be eight. The first slice, the second slice, the third slice, and the fourth slice. Okay, then we're going to use the midpoint to find the height of the slice. Okay, so you can see that if I send that around. What I get is a cylinder, and its volume is pi r squared H where h is, too. And then ours however much this is. Okay, so we have sliced Juan. It's following his pie 1.5 squared times h which is to slice, too. Okay, it's our looks, like about 2.2 five times, 2.2 square times too sliced. Three, 3.9 maybe pi 3.9 squared times, too. And then sliced for hi. Uh, maybe, like 3.1 times. Two. So then the total volume gonna add them all up? Okay. They all have a two and a pie. So factor that out. So I need 1.5 square, which is 2.25 plus 2.2 square, which is 4.84 plus 3.9 squared, which is 15.21 I think. Point on squared 15 point you one plus 3.1 squared 9.61 That's two pi times, uh, had those together for grant aid, or plus 2.25 31 91 times, two times by I get about 200. Okay, if that's not exactly what's in the back, remember? I was just estimating the height also, I just drew this. I didn't copy the one out of the book. Okay? So you might look at it more carefully if I didn't get the same answer. Okay, Now we're going to send around the y axis, and we're gonna make for, um, cuts or four slices, so we gotta ups. We're going from 0 to 4. So the width of each one is just gonna be one. So there's the first slides. There's the second slides and the third and the fourth. Okay, so if I send this one around. You can see that it's gonna be a big old disk with the hole in it. Okay, so this gap right here between here and here makes this whole. So to find the volume of that, you have to do pie. The big radius squared minus the small radius squared times to hide. So what you're doing is the very the biggest the big cylinder minus the little cylinder, which is the whole Okay, so this time we've gotta find this, which is big are Oh, sorry. Okay, I use the midpoint. This which is big, are in this, which is little are so slice, Juan. Okay, let's do sliced number big are little are and then r squared minus R squared. Sliced number one big are is 9.9 and little are is 2.2 sliced too big are 9.3. A little are are but there are 2.8 sliced three big are, uh, nine. I think the other one better be 9.5 because I'm going to give that 19.3 and little are is 45.3 and then sliced for big Are is nine and little are 6.5. Okay, We're not going to get the same answers before because we're making a totally different shape. This will look like the top half of Ah, if you cut a bagel in half, okay? I'll have a hole in the middle and be all weird shaped. Okay. Where's the other one? Looks sort of like a pair when you went around. All right, Now I'm going to dio r squared minus r squared. Okay, 9.9 squared minus 2.2 squared. So I get 93. So 93.17 95 cups 9.5 squared minus 2.8 squared 82.41 9.3 squared minus 5.3. Squared 58.4 and nine squared minus 6.5 squared 38.75 So the volume it's gonna be pi the some of these things times the width. And this time the width is only one instead of two. So 38.75 plus 58.4 plus 82.41 plus 93.17 to 72.66 times pi. I get it. But 857 for this one. Okay, fine.

Oklahoma State University

Applications of Integration