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"f' Which 9 "Jo (9~3y+ 1| (9-3y+ None of the +Ke-9) %1 7 line expression gives 4 4 above 7 4 Jdy Zx+V-2 2 and the volume the of the two coordinate so...

Question

"f' Which 9 "Jo (9~3y+ 1| (9-3y+ None of the +Ke-9) %1 7 line expression gives 4 4 above 7 4 Jdy Zx+V-2 2 and the volume the of the two coordinate solid axes generated around by rotating 3 line the first X=-2 quadrant region

"f' Which 9 "Jo (9~3y+ 1| (9-3y+ None of the +Ke-9) %1 7 line expression gives 4 4 above 7 4 Jdy Zx+V-2 2 and the volume the of the two coordinate solid axes generated around by rotating 3 line the first X=-2 quadrant region



Answers

In Exercises $7-14,$ use the Shell Method to compute the volume obtained by rotating the region enclosed by the graphs as indicated, about the y-axis. $$ y=\sqrt{x^{2}+9}, \quad y=0, \quad x=0, \quad x=4 $$

In this problem. We're taking the region bound by X y equals X squared and X equals two and the x axis. And so this triangular region, which will go ahead and shade in blue here and we're taking this region and revolving around the Y axis. And that's gonna form a series of washers or doughnuts here. And so we'll go ahead and start off with the volume is too. And then we're going to integrate from Well, before we even start that will have pie you gonna grow are square minus little r squared and in this case, be big circle that we have Or the big radiance is from the y axis outwards and that goes out taxes too. So the Big Radius is going to be, too, where minus and the small radius here is going to be, um, until we hit the first line there. And that's going to be when we set up things around the y access. Then our functions need to be in terms of why and so that's going to be minus square root of why spire so basically solving y equals X squared four x. We still have pie out front and then we're integrating. Our limits on this case are gonna be why values. And so the point of the origin, of course, at 00 And the point out here where they intersect is at 24 again, we're interested in why values, since we're evolving about why so that's going to be from zero up to four. Simplifying this we're going to get this is going to be, uh, the integral Well, it's just rewrite it high and a girl 04 of four minus Why and b Why and taking the anti derivative would be to these expressions lost all the constant of pie Morehouse. Or why minus y squared over to evaluated from zero four, substituting zero in get high 16 minus 16 divided by two. And then, of course, when we substitute subtract zero, it'll end up canceling those terms. So in the end, the volume created by all of these washers, added vertically from 0 to 4, will be a volume of a

You want to find the volume of the solids that's reforms revolving around the region. Help white to be equal to four plus two x minus X squared then y to be equal to four. My next words, X So we are looking for the volume about the given line. So we are looking at it in the X and start a horizontal to access. Then we look at it being the another plane as well. So, looking at this, we need to find the area of intersection where they intersect us. So looking for the intervals we create this and that's and if you equate, this is where you get so this becomes minus X square plus three is equal to zero. Let's look for X X is equal to zero and X is equal to three. So when X is equal to observe, why is four and when X is equal to three? Y is what one So they intersect at this point, this integration in that integral actual. So if I want to find about text and sees then my art So what? The x Isis? Yes, they are. Yes, uh, X. It seems to be equal to four plus two x mine It's my ex square and my small hour of eggs is going to be equal towards four minutes X Very good. So if I want to find a volume, my volume is going to be five Pain Sica from several 23 mhm of this Quigg minus Would it square? So I have four plus two x Yeah, my name X squared. Oh, swear before it's Oh, square e Thanks. So if you simplify Okay, lets simpler So you face what? I have my 18 ft Super pay as you go. This is a three. This is the castle. It's been a stripper for ex student for mix for No Yes, the barbecue PM blood. Okay, so let me clear that right? It's walk Mhm. Yeah. So I have five. Yeah. Yeah. Okay. Is okay. Okay. Yeah. So the XK 24? Yes. Yes. You find in which is in the same room is 20 high this if it is even simpler 500 by five. Is there any incident? Powerful. So it's my fault. This is plastic. What do you do? You survive Divided by B. It's cute as this is going to be one sided, but city provided by doing so, uh, anyone. So if you consider the Israel's that's still go there so you know, it's so if you you're getting 153 blighted by Yeah, okay, you can buy. It is the volume when you look at it in the expenses. Now let's consider if you are lying, it's we have a line equal to, so be mhm. Let's see a house. Me. It's when I want to find the volume such that it's a line a line. Why it's equal to one if why it equals 14 Happy kids. Let's write it out. Yeah. Mm hmm. Safe. Why he caught one. Then I can write my art. It's and uh huh. So I have Don't forget, I'm going to subtract one for me. So my r x, my big r X is going to be words my four flats in four minus one, minus this one. So this will give us what's going to be plus minutes with X ray. And then we want to do minus eight. So now let's fall falling. Be interior from zero. See? Mhm. Did he? So I have three. Let me be the S minus X squared. Okay, nine. Is this three? My late X Also the s. Yeah. So let's simplify it and see. Is he okay? This quake Meaning this quake? I didn't even Christmas too. Right? Okay. Okay. They're giving us. Thank you. From the thanks. Yes, extent of all. Thanks for here. Okay. Right. Four x to the power three knows me is Yeah, me x grade Blush magazine. He Yeah. Okay, so you have s to live my life. This is names even the powerful. Because there's so much for that. It's never for 9. 23. The power as to the power three. Uh, very much want to be provided by to wait until we have nine. It's going to be a is also to three. So they're only gonna enter which is going to be he traded by, please. Volume when the line Y is equal to one. Mhm

Yeah, eso if I'm reading this problem correctly, We're looking at the parabola, uh, to sideways travel of Michael's X squared. Um, and we want this area from here to here, and we're revolving around the y axis. And that's how I'm gonna answer this question and see if I get the same answer. Um, now, because we're revolving around the y axis I want in terms of why so d y here? So when they gave me that, um, would they give me Why equals X squared? Okay, I need to stop myself, because that I drew this wrong y equals X squared should look like this. This is the area we're looking for. Um, just so when I see that though, I wanted to be X equals So it's a square root of why, um, as one of my functions Now, they were nice and they didn't give me my other function in terms of, you know, X equals two, which is not why. So we're good here. Eso there's definitely space between the access I'm a revolving around and what I'm revolving. So that's definitely a clue that we are doing the washer method, which is pi times the function That's further soy, which is X equals two squared minus the function that's closer, the square root of why being squared. Um, now the other thing is, everything has to be in terms of Why. So what is this value? What's the Y value? This one's clearly zero. I think that's obvious. Um, but what you'd want to do for your upper function is actually figure out what the Y value is. When excess two squared, we'll give you four. Um, so let's just clean this up for a little bit before actually solving it. Um, in a row from 0 to 402 squared is four square root squared will cancel. Such as Why do you Why so now do the anti derivative, which would be four y minus one half y squared. And that's from 0 to 4. So it's nice about this is you're just plugging in four. And for your wise four squared is 16. Half of that is eight. Plugging in zero will not change anything is four times 000 squared to zero subtracting that won't do anything. So 60 miles eight is eight. Pi is your final answer

So let's get the graph here, and we will use the washer method. So there's my elemental washer, and the upper radius are or the bigger radius are equal before X and the smaller radius. Lower case R with people to X. Q. So the elemental volume devi will be pi times r squared minus r squared DX. So now we can apply the limits for the integrated. So that would be the girl from zero to two and a big kiss. Our is four x squared minus X cubed squared DX. And when you do this, any girl, the integral comes out to be equal to 512 over 21 then type that with pies with the volume generator would be 512 or 21 pie. So now the axis of rotation is the line bicycle to eight, which means all distances would be measured from that line and again use the washer method and for the washer method, the outer radius, our would be equal to it, minus X cute and the inner radius. Lower case R would be eight minus four, so my Devi will be equal to pi times. Eight minus X cute squared minus four minus Sorry. Eight minus four X squared dx. And then in order to find the volume, I just need to apply the limit from 0 to 2. So it minus X cute square minus eight minus four x squared dx. And when I find the integral, the integral is 832 over 21 times it would pipe. So the volume is 8 30 through over 21 times pi.


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