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(1Polni) Use tha dlsk mathod or the shell methad to fnd the Volumes of the sollds generated bY revolvlng tha region bounded the graphs of the equatlons about the gi...

Question

(1Polni) Use tha dlsk mathod or the shell methad to fnd the Volumes of the sollds generated bY revolvlng tha region bounded the graphs of the equatlons about the given Ines TXa) tha {-axls volume cublc units.b) tne Y-axis volume cublc units.

(1Polni) Use tha dlsk mathod or the shell methad to fnd the Volumes of the sollds generated bY revolvlng tha region bounded the graphs of the equatlons about the given Ines TX a) tha {-axls volume cublc units. b) tne Y-axis volume cublc units.



Answers

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the $y$-axis.
$y=x^{2}, \quad y=2-x, \quad x=0,$ for $x \geq 0$

For problem 11. We need to find the volume if this area was revolved about the Y axis. And so if we set this up using the shell methods, so v equals two pi integral from a to B of some radius time, some height DX, uh, we can go ahead and start to apply this to this problem. So once again, we'll have two pi times the integral, and we're gonna be making little slices of this shape. So we're gonna have this height, and they're gonna have this distance here are on that distance is gonna be just the distance from the UAE access, which is gonna be X at any given point. So we could make x r r Multiply that by the height, which is gonna be this upper limit of the square root of X minus the lower limit of two X minus one to minus two x minus one. So if we clean this up equals two pi Oh, we also need to add our limits of integration, which you're going to go from 01 and one is where these intersect, right? If we plug one into both of these two times one minus one is gonna equal one. The squared one equals one. Right, So that's where they intersect. So two pi times integral from 01 on. We can go ahead and clean this up so x times and we're gonna make this extra one half power minus two x then plus one, right. Those negatives are gonna flip each other d x And finally, let's distribute this Xa two pi to grow from 01 x times X to the one half is gonna become extra three half power minus two x squared plus x then TX So now we have our integral set up. We can go ahead and actually solve it. So that's gonna give us two pi. So we're gonna add one to that power that we come. Five House five house Divide by that two fists minus. I'd went to this power so execute divide by that two thirds plus one half x squared from 01 Here we go. So now we just need to plug in our limits of integration, which will give us two pi. It's two fists minus two thirds plus one half, right. We just plugged in. Want all of this she plug zero into all of this? Well, it well, I'll just go to zero, so we don't have to worry about any of that. So if we plug this into our calculator here to fifth minus two thirds minus one half or sorry, plus one half, that will give us 7/30 which, if we multiply by two, will give us 7/15. So 7/15 pie, which is going to be our final volume.

For problem 10. We need to find the volume if this area here was revolved about the y axis. So to do this, we're gonna use the shell methods of two pi times integral from a to B of the radius height, the X s. So if we apply this to this problem, we're gonna have two pi times the integral and we're gonna be making slices little vertical slices of our area. And then they're gonna have this height, and they're gonna have a radius here, which is gonna be the distance from the Y axis. And that distance is just gonna be X so we could make X r r We're gonna have to multiply that by the height, which are height is gonna be two minus x squared. So to minus X squared, we're gonna have to subtract are lower limit, which is just gonna be X squared. So minus X squared D X and then our limits of integration are gonna go from zero. They're going to stop where these two intersect at one. So that will equal to pi to girl from 01 We're gonna go ahead and clean this up so negative X squared minus X squared is gonna become negative. Two X squared is gonna have to x minus two x squared except Times X is gonna become execute the X and so we can actually go ahead and factor out a two here. So if we pull that out front, that will give us four pi z integral from 01 of X minus X cubed. Actually, we're not going to do that. And I'll show you why in just a second here. So if we just go ahead and integrate at this step, that will give us two pi x squared and then you divide by two. So that will just become X squared. And we add one to this powers that will become minus X to the fourth and then two divided by four is one half and from 01 So now we can go ahead and apply these limits of integration. So I will give us two pi and we plug in once that will give us one minus plug in one again one half. And if you plug in zero, both of these will go to zero so we can ignore that step. So then we will be left with one half times two pi, which is just gonna be pi. So pie is going to be our volume.

Okay. What we want to do is walk through the process of being able to determine the volume of the region banded by X. Equal to y squared. Why go to one X. Equals zero about the X. Axis. And we're using the shell method. Um And so the first thing we should always do is to draw a graph of our region and then label things and so forth. So if this is X. And this is why Xing otherwise squared is if why is one X. Is one. So it really actually looks something like uh huh. Something like this. Um And if I if it's easier for you to kind of think of what I call the square root of X. That's what it would look like. Um And we're going up here at one as well, whoops forgot I was in eraser Road. And then this is one, right? So this is a little region that we're interested in and we're evolving it about the ex taxes. Um And so first of all we want to do then draw representative rectangle. And so here would be our representative rectangle where this is delta Y. Um the height or the length of the rectangle of course is why squared and the distance from the axis of rotation is why. Um And so there we there we have him. Um And so now um delta V is equal to two pi the distance from the axis of rotation. Which is why times the heights of the right tangle, times the width of the rectangle. And so this is going to be equal to two pi. Why cubed delta Y. So the or the volume is equal to the integral. And in the y direction we're going from 0 to 1 of two pi. I'm gonna bring that to pie out of Y cubed and delta Y. Becomes a. T. Y. Okay so now we know how to integrate white cubes. So this is going to be actually 1/4. So this would be um pi over two times y to the fourth and we're going to evaluate it at one and at zero. So this is just going to be equal to pi over two is our volume.

Okay. What we want to do is we want to find the volume of a solid generated um about the Y axis of the following region. And the first thing we need to do is to kind of get that sketch developed. Um and just to kind of help us out. And so the first thing we do is schedule region and so this is one, this is zero um and appear at one. And so we know, oops that went off at smart, didn't it? Um We know it looks something like this, that is why you go to exclude. And so we know it's why equal to zero. So it's going to be the X. Axis and X equal to one. So it's gonna be this region right in here that we're evolving about the Y axis. Okay. And of course we're gonna be doing um the shell method. So let's get our representative rectangle drawn. So let's just put a rectangle right here. Um We know the with that rectangle is delta X. The heights of that rectangle is why equal is X squared. And the height of the rectangle of course is is to this X squared. And then this distance to the center of the right tangle is X. Right? Um and so we know that the changing the volume is actually equal to two pi X. So it's gonna be this distance right here, two pi um X times the height time sir with that rectangle which is about two pi X cubed delta X. Right? So here is our representative volume and so now the volume is equal to the integral and we're going from 0 to 1 and this will be two pi X. Hope that should be a cubed dx and there we have them so now we can kind of integrate and find that volume. Um bring out the two pi and then of course this is going to be 1/4 x to the fourth. And we're gonna evaluate at zero and one. So this is gonna be pi over two times one minus zero. So this is just gonna be pi over two. So there is a volume right there.


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