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(Hpu) Let Rbe Ane region in the lirst quadrant betuen Ol tlie right by the curi? lulz) (See picture )HudAlld botutuedlUsing the Dbsc Washer Method, set "p (Im...

Question

(Hpu) Let Rbe Ane region in the lirst quadrant betuen Ol tlie right by the curi? lulz) (See picture )HudAlld botutuedlUsing the Dbsc Washer Method, set "p (Imt do NOT evaluate] the integrall - peexlee t €npuie the' rolutue ofthe solial fonex wlicn rctating around IUaing the Shell Method, st Up (but do NOT evaluate} the integral(-} nexrxlevl compte tlie volute &f the solid forinesl when rotating R around the y-axis.

(Hpu) Let Rbe Ane region in the lirst quadrant betuen Ol tlie right by the curi? lulz) (See picture ) Hud Alld botutuedl Using the Dbsc Washer Method, set "p (Imt do NOT evaluate] the integrall - peexlee t €npuie the' rolutue ofthe solial fonex wlicn rctating around I Uaing the Shell Method, st Up (but do NOT evaluate} the integral(-} nexrxlevl compte tlie volute &f the solid forinesl when rotating R around the y-axis.



Answers

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the $y$ -axis. $$ y=x^{2}, \quad y=4 x-x^{2} $$

Okay, we're going from X equals 02, X equals one from Michael zero to Y equals this curve which is like each of the minus X squared which is each of the minus X squared over two times a constant. So when x zero we're at that constant, let's just put it right here. When X is one were lower than that. So we just got this little thing right here, let's say. All right, so we're going to send this around the y axis and then we're gonna find the volume using shells so we're gonna cut vertically. So here's a shell. Okay? Volume of a shelf two pi R H T. He is the thickness of the shell. So in this case that's D X r is the distance from the axis of rotation? Out to the shell which is X. H. Is the height of the show. Which is why on the top minus y. In the bottom. So one over square to two pi each of the minus X squared over two. Okay. Still growing it. Find the volume by doing two pi integral x times one over square to two pi E to the minus X squared over two D. X. So we have this constant, we can bring it out. Oops, I forgot to say from 0 to 1 and we have E to the minus X squared over two times X dx. I'm gonna let you be minus one half X squared. So do you is minus one half times two X. So minus X dx I need a negative sign here which puts one out here. Okay? So you get minus the square to two pi when X zero U. Is zero. When X is one U. Is minus one half E. To the you do you get minus the square to two pi in a girl V. To the U. Is E. To the U. So each of the you from 0 -1 half, so minus square to two pi E to the minus one half minus E to the zero. All right. So I'm just gonna make that positive to pry square two by one minus one over the square root of the E.

I'm gonna assume from this problem, this four minus X squared. Oh and y equals zero. So we're talking about this region in here. But what I'm actually gonna do, so we have y equals four minus X squared is I'm gonna figure out where that intersects The x axis is set equal to zero. And When you square root you do get positive negative two. But if my understanding of this problem is correct, as you revolve using the shell method, What you can do is just consider from 0-2 because as I revolve, it's just going to hit the same points over and over again. So the show method is two pi from 0 to 2. As I mentioned, of the radius of each one of those cylinders, I don't know if you can tell it's a cylinder inside of there. So each radius X. Units. Um And then the height of each cylinder is a 4 -1 sq dx. So it would be smart I think to simplify before doing the integral. So distribute your four into your side of the X. Into the problem Still D. X. So from here do the integral where you add one to the exponents And then divide by your new, expanded to four divided by two is 2 And then add one to the exponents. Multiply by the reciprocal and we're doing this from 0 to 2. And as I plug in twos or two squared is four times this too would be eight. And then 2 to the 4th power. Well how about this? Because two times two is four would cancel with this. So I can just do two squared is four and plugging in a few zeros will not change the value of those. Um and again, don't forget about that two pi front. So anyway, 8 -4 is four times 2 will give me eight pi. Yeah.

If you look at this problem, why goes X squared plus four and you can see that they just care about the right side. Uh And you can double check That the upper functions y equals eight. Um And I should also write out my goes X squared plus four because um well if you set them equal to each other let's do that Equal to eight. And then you start to solve this by subtracting forward to the right side. 8 -4 is four. And then when you square root both sides you get x equals two. I know it's possible negative too. But all I really care about is how to come up with this two pi R H equation and it's two pi the integral from 0 to 2 because we're doing each one of these pieces and yeah the r value is just X because that's the width of each cylinder that we are revolving around this axis. But the height is the upper function of eight minus the lower function of X squared plus four dx. So a few things is I would actually simplify you know by distributing us in here And then 8 -4 will be four. But you also want to distribute this X into it. So it's four X minus x cubed. Um You know after I distribute the accent to the problem Still from 0 to 2 and from here I would be able to evaluate the integral Adding 1 to the exponent and dividing by your new excellent four divided by two is 2 and then 1/4 X to the fourth Going from 0 to 2. Still it's a two pi in front. What's nice about this is if you plug into in for both those X is two squared is four times two is eight, two to the fourth power to for a 16, divided by four before and plugging in. Zero is you're just going to get a couple of zeros. So 8 -4 is four Times This two would give me eight high.

The key, there's a few keys of this problem is the x squared function of something like this. I don't know exactly what the y equals four X minus x squared is but what we need to do is figure out where they intersect each other and yeah, so where do they we can just set them equal to each other. So four x minus x squared is equal to x squared. And what I would probably do is add x squared to the right side, so be two X squared and then minus four X. So from here I could actually factor out a two X or left with x minus two there. And the zero product property would say either two X equals zero, so expense equals zero or X could equal to. And then we can set up our own and rule. Uh using the show method is two pi and each one of the radi I um trying to think of what each radius would be. I guess what we're doing is uh like each radius would be X. And then you just have the upper function, which is that four x minus X squared minus. The other experts have been minus two X squared D. S are going from 0 to 2. Uh So it would probably be worth your while to simplify that. The reason why I say that is so you make sure that you write for x squared minus to execute dx so you don't mess up your integral where you're adding one to the exponents and then you multiply by the reciprocal of your new exponent or divide by your new export. Whatever works for you And two divided by four reduces 1 half. We're doing this from 0-2. So if you plug in two and for your bounds um Plug in your bounds, I guess two cubed is eight times at four would be 32/3. And then to, well one of the tools are canceled here. So just to Cuba which would be eight Um and I don't have to worry about zero because plugging in zero would give me a few zeros there. But if you were to get the same denominator that would be like 24/3. So 32 3rd minus 24 3rd would equal eight thirds times this two gives me 16 3rd. And don't forget about your pie should be the right answer.


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