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Problem 66 Hard Difficulty

Volume of a cup $A \in$ -inch-tall plastic cup is shaped like a surface obtained by rotating a line segment in the first quadrant about the $x$ -axis. Given that the radius of the base of the cup is 1 inch, the radius of the top of the cup is 2 inches, and the cup is filled to the brim with water, use integration to approximate the volume of the water in the cup.

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this problem. IHSAA is enveloped by light, probably need to kind of work backwards. And in this case, what they told us we need were given a case where we're told that the soul vis a solid that looked like a plastic cup is generated by rotating a certain ah, certain co. But a line order, that is, it's really pick it out in the first quarter. Okay, out there, it's There's something I would hear the form that went notated about the X axis produces a cup, a plastic cup like figure. Okay, And whose three out of operating is givens refer to draw the cup. Ah, like Well, let's say that's a cup. We're told that we produced this sort of this produces, shaped like a plastic cup, is produced with the the radius. Up be to just and really is the bottom being one image. That's the home. So remember that the very two and then believed, discovers filled with water, and we need to find out the volume on Remember that, Uh, let's see. The key here is your total, its location about the X axis and so that we know all along that that's a case. Uh, the integration is gonna proceed along the X. Okay. Which means we have a from something like we're looking at, you know, have toe. Um, look for this Libya seeking this kind off son. Okay. And this is a good excuse it just because the solid objects of the official rethinking toe the disc problem, which is this s o d x right where this serves as the radius left affects else a serious? Um no. Which means I'm really working backwards here. So radius would look, that's a case radius. That's it. Looks well, the white Derek do something like this. My next. That's all minerals. And, you know, it changes be the decreases or increases. And then we can bacon kind off constructing backwards. If this is the sole coldest, a little, you know, that prejudices that direction so we don't have to invert the cup pressing the America Cup. Okay, Ready, Lord, the cup. You could look like this. Okay. Right. And so that and that will be the radius. So be almost there. But we need to find basically, we're looking for a function like that. The first quarter have been told. So we're looking for? Ah, line a linear line like that. Uh, and which will rotated about the X axis produces Review. Sissy, uh, shaped like this. Okay, so we're almost there. That's the radius. And but we have a lot. Except we don't know what the lines. What the What is a form of line? This is therefore picks. But how does it look like it? Looking useful information that's going to us. We're told this. We're told the upper radius is two inches. The bottom releases one. And we also told us the high tous Hi. This six. It told the hype this. So you gonna map this figure here? And we can always construct conveniently. I'm struck. Position the base on the bottom to lie along the way. Access? Why not? Right. And so we have. This will be had that form rotated about the the X axis. We know how a plastic about the cup off of your city, Right? Okay, So orders are we told. Let's that's important information. The bottom radius is one INGE, it just This is the bottom radio. That's that's one one. Okay. And the top radius is two inches. So this distance here would be two inches the highest, too, that we don't x coordinate what we do, right? Because you know the person a couple of good, that's a base. The height is given a six inch. So that six to what kind of looking backwards. And this would be zero. What? Okay, so now do we You know, the function, the straight right. You have to, Gordon, in your love, those values. And and they have when you find the slope and then the awful back. That's what we should do next. Well, I'm going to now figure out so y equals MX. See, they get out. Question And and then, uh, Quentin Volume lugging it back in that interview. This problem. So what is the perfect little one? 62? We have two points. Would be to minus one one, divided by 6.0. Book us. Well, let's see. Slow. You have to point little one and six to, So your m is. You know, U minus one, divided by six minutes. 0 1/6 Okay, then let's plug in this value here, for example. Well, we have one equals 1/6 times excess little. Let's see, this means C equals one. We know em, you know? See, you have a question on the line by why once 1/6 6 of one. Okay. And this is it's good to keep it this week just like that, because you can have to do integration that respect to extra Find the volume. The point of William we just now we have a perfect and then is known bees know because is it'll be a six. My, this is 00 here and then a six year for a beat on a good eight ago. So but the help create the volume volume It's five times you know, zero to six if a big squid, but you can do it expanded. Or next to that on one of the next six of the world's greatest X squared over 36 plus one plus by status excellently the X Once you take this but look like I I'm 0 to 6 x cubed Over three, you will become statuses plus X well x squared over. It seems to with zero second vanish like in the value of six l quick excuse to 16 rented by one week less excess. Six less. Explain this to the 6/6, which is by times not exactly two. Bless six plus in exactly six. We have 14 times bike and the highest genital was seven. You get on a volume of water. That can be what in? From the couple. What for numbers All worker really want.