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Solid generated by revolving the region enclosed by the curves 09 Find volume of the y = 32x and y = 2x2 about axis...

Question

Solid generated by revolving the region enclosed by the curves 09 Find volume of the y = 32x and y = 2x2 about axis

solid generated by revolving the region enclosed by the curves 09 Find volume of the y = 32x and y = 2x2 about axis



Answers

Find the volume of the solid generated by revolving the region $R$ bounded by the curves $x=\sqrt{y}$ and $x=y^{3} / 32$ about the $x$ -axis.

Here Cushion is if c ah, the problems of elation revolving about, uh by axis. Third is y axis and the problem is revolving about bikes. So grab, it will be like this. And if we take on a strip, this is X And this part is the way sub volume generation will be always by excess Square De Wei in Dickerson No, we have given vie Quito X by who have minus and I dont exquisite. So squaring both side by square will be extra square minus four minus x square. So final part full rice square minus y squared X x squared equal to excess Quiet So fellow up excess square will be Colin Rice Square It could tow excl Ever take common one plus y squared Hence excess square will be full by square by one place by school No put this value in the graph It is by excess square d y. So bottom generation will be I will be outside full also will become our side by school by one place Why skirt the way and ah learn a prelim it'll be 0 to 1 So finally it is full of I And if we add one and subject one, then question like this. So this is four by and 0 to 1 one plus vice square by one place by square minus, derided by one plus y squared Geritol one. So finally, if you organize this so full by and do I will be by someone minus down in verse. Why? And it is here Cuban. So finally it is hold by one minus 10 was one minutes by very food. So final answer is pull if I and go full minus pi buy food.

Eso you don't need a graph to be able to figure out this problem because one of the lines that its revolved around is, like all zero nine minus x squared. So what you need to do is because they both equal why you want to find the point of intersection by setting them both equal to zero. Um, so probably the best way for me to do this is to square both sides. Um, And when you square both sides, uh ah. I guess the square root would cancel that and then on the right side of equal zero. So we're looking at these are X equals zero. And then these would be, um nine would equal X squared. So actually be plus or minus three. Well, it's her minus three. Um, now you just want to double check that the one of these answers is correct, because when you square, you might get extraneous solutions. So what that means is, if I plug in three into this problem, um, I need to make sure I get zero. Well, if I plug in three for this X and the sex, I do get three times zero, which is zero and then same thing with the other. Um, what I'm going to do first is just do the positive values. Mhm. So I'm just going to go from 0 to 3. This is still the disk method, because we're evolving around the axis, the X axis and Michael zero is that X axis. So you can just do X and then the square root of nine minus six. And again, we actually did this earlier, but I would actually foil this out. Nine x squared minus. Um, thanks to the fourth, it's a pie from 0 to 3 of that. Okay, So add one to your exponents. Divide by your new exponents, minus 1/5 X to the fifth, and that's from 0 to 3. So as I plug that in, uh, well, I know that this is 81 but I don't know what three to the third power to the fifth power. Excuse me is it's 243. Yes, as I simplify that, anyone minus 243 fists is about 162 fists. Now, don't forget about this pie. Over here is Well, now, here's the deal. Um, is we're missing half of our answer. Because if you were to look at the graph of this, um, we only did from 0 to 3, and we got this positive area, but we're also missing from negative three negative 3 to 0, which hasn't more positive area. Um, so what? You what I should have done to begin the problem is realized that that even function needs to be doubled for all of these. So the actual answer is double that to get 325 pi over five. Sorry, I'm miss saw 324 pi over five.

In this problem. We're trying to find him of all of you, of the solid phone, by retreating the bounded area in the graft bill about the white activists. No, since we're going to be doing X integration and were involved about vertical axis than it appears that we will be using the shell method. So let's first identify oh, samples the lender reading the equal to act under Hank equal to you Need to negative. You're wrong with being 42 times the inter, both from X equal zero flexible one one x time. You need to know the best square. Yeah, No one took danger, bro. We have negative one house dreams you need to the negative range thing. While you did it to one that would be have bring out the negative Bahnhof Hi times. I need to the negative born minus into zero that would be equal to one minus. One of it in temps way is beyond value

So for this problem we are going to be finding um yeah if we have the region bounded by, why equals E. To the X. So I'm just going to draw a sketch here. Um Why equals E. To the X. And Y equals zero? Which is um the X. I'm sorry why zero is the x axis X equals zero is the y axis and X equals Ln of three. So X equals Ln three is just gonna be a vertical line. It's just a constant whatever Ellen of three is. And we are rotating this area that's founded by all four of these things were rotating that about the X axis. So then my other side will look something like this. So we are going to have to use the disk method here. There is no empty space in between. So we don't have the washer method, it's going to be the disk method whereas the width of each disk is dX so that's going to be, the volume is equal to um Pie times. The integral From your first bound which is zero To your second bound is Ellen of three. Because those are our X. Value our bounces from zero to L N. F. Three. And it's going to be r squared. So your radius squared times the width of your disk which is dx. So then we need a representation of the radius. Well the radius is from the center to eat the X. So our radius is E. To the X. So our volume is pi times and a girl from 0 to 1 of three of E. To the two X. Dx. And I can solve this by hand. That's pi times the anti derivative of E. To the two X. Would be one half E. To the two X. And you can double check that by knowing that the derivative of two X. Is 22 times a half is one. And then the derivative of E. To the something is still lead to the something. So we need that one half in front to counterbalance the two here And we're going from 0 to Ellen of three. So that's pi times 1/2 E. to the to l. N. minus one half E. To the two times zero, which is just zero. Yeah. Alright, common mistake here is that students will try to cancel E. And the L. N. Right away. But you actually need to use um the power rule for logs and move this to the three so that it becomes E. To the L. N. Of three squared minus one half. Another. Common mistake either. The zero is not zero either, the zero is one and then I get pi times one half now I can cancel the E. And the Ellen. They can undo each other um Once that too is part of the three instead, so be one half of three squared which is nine minus a half. So you get pi times nine halves minus a half is eight halves which is four. So my answer is four pi. The units cute.


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