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Question

Find the volume of the described solid $ S $.
The base of $ S $ is a circular disk with radius $ r $. Parallel cross sections perpendicular to the base are squares.

Wen Zheng · Accepted Answer

the problem is finding the volume of the described the solid as base of ice is a circular disk with writers are her lower cross sections of articulated the base A squire&#x27;s with this problem for any cross section When dick cooler to the base. It is a squire here. I think note area of the cross section is the ax. Your axe. Is that this tense? Between original point here zero two cross section. So first to winning to compute the side of the cross section. So here, if this is our this party&#x27;s harmed Linus Sacks with a side of cross section is our squire minus. I&#x27;m minus x square. You took this one once applied to this&#x27;s the side of cross section the axe. This is called or I&#x27;m sorry, squire. Minus harmony scored that. This It is equal to four to our backs minus tax square. That&#x27;s a volume of the Sorry. The ice is the coat you into girl from zero to two are max yaks. This leads to the integral from zero to two are or to our backs minus squared. Thanks. The answer is sixteen over three times. Uh, cube

Carson Merrill · Answer

So for this problem, we have a circle with radius R, and it&#x27;s centered at the origin. So we have is X squared, plus y squared equals r squared. Um, and we solve for y and we see that y equals plus or minus the square root of R squared minus X squared. So we know that the top half will be the positive square root. Um, and the bottom half will be the negative square root. So the base that we have is two times r squared, minus x squared, um, and that will give us the whole base length of RSO sleeves triangle, Then the height is the same for every triangle. So the area of one cross section is going to be FX equal to one half our base, which is to route R squared minus X squared. Times are height, which is going to equal our height, times the square root. Hi, because the tubes will cancel. So just high times square r squared my sex work. Then we know the circle intersects at the X. The X axis at X equals negative. R and X equals positive are so that&#x27;s how we&#x27;re going to integrate we can pull out the H, and we also know that it&#x27;s symmetric. So rather than going from negative, are are we can just go from zero to our and multiply it by two. So our volume is going to equal two times h times the interval from zero to our of the R squared minus X squared G x then for part B. Um, what we find is using this. We can take the we can square both sides. We get our equation of a circle again. Um so since Y equals the square root of X r squared minus X squared, that&#x27;s our upper half of the circle of radius R centered at the origin. So the way that we can write this now is going to be the volume once again, up to age, we have that same thing. But that&#x27;s actually gonna end up equaling two h times. One fourth of pi r squared. So then the final answer will be one half pi r squared H and we were able to do this by replacing the integral with the area of a circle which we know to be pi r squared. And then we multiply that by 1/4. Since we have, um, half of the half circle

Chris Trentman · Answer

were given a solid S. And we&#x27;re asked to find the volume of this solid Dude, we&#x27;re told the base of S. Is the region enclosed by the curves? Y equals two minus X squared. And the X axis on the tip of the chain wallet chain. And we&#x27;re told that the cross sections perpendicular to the y axis are quarter circles. Listen, so first you want to find the volume. We need to find the area of each of these quarter circles. To find the area of the quarter circles need to find the radius of them to find the radius. We need to find the distance between the left and right sides of our region. Well, solving for X, we get that X is equal to plus or minus the square root of two minus Y. And so therefore we have the radius go of each of our quarter circles. I&#x27;m sorry, is actually going to be using symmetry two times the square root of two minus Y. Didn&#x27;t have to say. And it&#x27;s adam do you said it? Mhm. And therefore we have that the area of the quarter circle. Well this is 1/4 of pi r squared, Which is 1 4th times pi times two times route to minus Y squared. This is the same as two -Y. Ooh some times pi mm somebody goes now, since this is the area of the quarter circle as a function of why we have their volume. This is the integral over all these quarter circles. So we see why ranges from zero up to the vertex of the parabola too. And this is integral from 0 to 2 of a of Y DY, Which is pi times the integral from 0 to 2 of two -Y. Do you why evaluating this integral? We get two pi. This is our answer.

Ma. Theresa Alin · Answer

suppose you want to find the volume of a right circular cone with height H. And base radius R. To do this we draw on the partition plane R. Right triangle with high H. And base radius R. Now looking at this, we see that the triangle is made up of the line. Y. Equal to H. And this line whose points are this would be Rh And this will be the origin 00. And so the lion formed by these two points would be why equal to that&#x27;s a rice overrun of H. Over our. And then times X. Now if we rotate this right triangle about the why access can then form the right circular cone. And so using a strip that is parallel to the axis of rotation, the solid form by rotating this trip about the Y axis, it&#x27;s a cylinder with radius equal to X. And height. That is the difference between the top function H. And the bottom function H. Over our X. And so by cylindrical shell method we have V equal to two pi times the integral from 0 to our. Since our area is found over the X interval zero to our and then you have radius which is X times height which is H minus H over our X. And then dx. And then simplifying we have two pi Integral from 0 to our of X. H minus H. Over our X squared dx. And so integrating with respect to X we have two pi times We have aged times x squared over two minus H. Over our times X cube over three. This evaluated from 0 to our So we have two pi times When exes are we have age times R squared over two minus H over our times are cube over three and when X zero everything becomes zero. So from here we have two pi times each over two R squared minus H over three R squared. And this is just two pi times each over six times R squared. Or this is just by over three H times R squared. Therefore the volume of the right circular cone is by over three R squared times each.

Rebecca Polasek · Answer

so I want to know the volume. If I have a right circular cone with heightened based well right circular cone were doing this shape and then you have a right angle here if you know what the straight up and down. It&#x27;s like one of those little paper cups that&#x27;s connected down at the bottom. And as you want to find this volume, you need that height, which is the straight up in down line. You also need the radius of the base so the radius of the bases are Remember. It&#x27;s just going from the center of that circle base, out to the edge, out to the perimeter. So volume for a right circular cone is 1/3 high R squared H. So that&#x27;s the set up. I wanted to write for some undefined figure at this point, not yet knowing Agent are, but being able to then in future, fill it in

Linh Vu · Answer

So let&#x27;s let you drown a craft on the place. Now, you know, the base will be this square. I&#x27;m just shop. Okay. 11 this 1111 We noticed that on by symmetry, we can do the volume technical. Uh, Jules, terms of the because mission will be the construction will be arm about decorating the want access. Which image? Consider the tough, but here and the other time. Ha ha. But I will be the same regiment in time by indication from the 21 Do you want? Only. Okay. And now jump with beautifying what&#x27;s inside Just in the ground here. You need to do that. We need, you know, Just be a question. Yeah, would be express 1 to 1. I understand. On this life hand will be minus Thanks. Was wanting what you want Children means then will be ex ego too. Well, I&#x27;m honest one in here. Thanks. Echo to one minus y. So the Greenland and we&#x27;ll hand ah men off the one minus one minus. Well, I&#x27;m just one. So you could shoot humanist. Why doesn&#x27;t will be the total, uh, diameter offered us to Mexico. So the area on the construction under serious. They&#x27;re going here, Jaha, By our square. So we&#x27;ll be one minus y square. So we&#x27;re ready to from just grab. Can Wouldn&#x27;t be. Uh huh. Yeah. Here. Uh, bye. One run. It&#x27;s why Square? So we&#x27;re equal to integrate from 0 to 1. You wish it had God, Joe? Uh, by outside. When my wife square. Why? So is it true? Did you equal to 11 of you? Hey, uh, you got your woman? Why do you go to my house? Ah, dig. Why? So when you could deserve you? Will you go to one? Meanwhile, I got you one. The new income pictured. Zero. So what happened? You into grown? Now we&#x27;ll be from 1 to 0. You square my enough Do you? You get You know, Mr Guy you much and you&#x27;ve got three going 1 to 0. So we&#x27;ve got people Jew by over three. That&#x27;s a nice

Wen Zheng · Answer

in order to find the volume we know we&#x27;re using the formula from negative art are a of ox d ox formula specified in the textbook. Which means we know we have your order. A of axes are in. Max is equivalent to the square root of R squared, minus X squared. Plus it&#x27;s word of R squared minus X squared. And this whole thing is squared. Now, you should have learned the foil method, which means we can actually expand this. When you square square root, it just cancels off all the stuff on the outside and what&#x27;s left is on the inside. So in other words, this could be written as four times r squared minus X squared. Okay, this is our backs. Which means we can now plug in again. Four can be pulled out. We&#x27;ve learned this before, in previous chapters that Constance can be pulled out. Okay, Now, we&#x27;ve also learned that we can actually split up into girls. We have two parts. We can split up into girls and write this with separate notations. Now, given this we know we cannot integrate, which means we can use the power rule, increase the exponent by one divide by the new exponents. We have X cubed over three instead of X squared, for example, substitute in our bounds. That&#x27;s the next stop, and we end up with 16 over three are cute.

Wen Zheng · Answer

it seems like you guys fell asleep so I might want were given a solid S and rest to find the volume of this solid, we&#x27;re told that the base of S. Is a circular disk with a radius. Little are that parallel cross sections perpendicular to the base? Are squares look well. But is it so the equation of the semicircle above the X axis? This is why equals the positive square root of little. R squared minus x squared. And the equation of the semicircle below the X axis is Y equals negative square root of R squared minus x squared. These bounds the base and therefore we have at the side length of the bottom of the square, which I&#x27;ll call S. This is our upper value of why? Which is the square root of R squared minus X squared minus the lower value of why? Which is the opposite of the square root of r squared minus x squared which is two times the square root of r squared minus x squared. Right? Yeah, all turns out so this is the length uh side length of the square pointing out of the circle. Yeah. Now define the volume. We should find the area of one of these squares as a function of X. Well, I was well this is the area of a square with a side length and so it&#x27;s going to be the side length two times the square root of r squared minus x squared squared, which is four times r squared minus x squared Israel to. And now we have the volume using the cross section interpretation. This is the integral from X equals negative art positive our of our area A X of the squares pointing out of the circle, D X. This is the same as four times the integral from negative are positive. Are of r squared minus x squared. The X. I can imagine this is a pretty simple integral to evaluate but I&#x27;ll walk through since we do have some and all the variables in this. So this is equal to take the pussy on four times R squared x minus well mr Niva. And this is X. To the just letting us know that Science third power over three from X equals negative are two positive are and if you plug in well we get four times To r cubed over three. It&#x27;s named that wasn&#x27;t there bitch? It rode a horse naked lady which is equal to she really she&#x27;s not very good doing china. He rides around yes minus four times negative. Two. R cubed over three. You can see it. This is equal to is sloppy eight plus eight or 16. R cubed over three. Good job. Really good.

Problem

Find the volume of the described solid $ S $. Th…

Problem 54 Hard Difficulty

Find the volume of the described solid $ S $.
The base of $ S $ is a circular disk with radius $ r $. Parallel cross sections perpendicular to the base are squares.

Answer

$\frac{16 r^{3}}{3}$

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