find-the-volume-of-the-solid-with-the-given-base-and-cross-sections-the-base-is-the-unit-circle-x2y2

Question

Find the volume of the solid with the given base and cross sections.
The base is the unit circle $x^{2}+y^{2}=1,$ and the cross sections perpendicular to the $x$-axis are triangles whose height and base are equal.

Linh Vu · Accepted Answer

So we have the base equal to I know you, Miss Circle X crab, I swear squatting. Could you won on the Conn Sectional? Maybe, but many correcting Thanks. Assist on with Chandler with their base. Go to her. And so let&#x27;s head to draw the picture in the X&#x27;s and y&#x27;s gland. Here. Next. Here. Why? Yeah, we have a unit Sicko. So we have. Yeah. And the constitutional would be on my medical to the X, and we have now. And now we were to compute the the land. I&#x27;m the cream line hand. We need to know the formula fund. A number of top pair will be. Why it could you just scroll in the one minus x crap? This one And here we had a phone. Now hear why I ever took in my nest. Screwed over the one minus X square. So the home hide here. You had a former, uh uh, the top man at the bottom. So we shouldn&#x27;t have thio one minus x square. So now we should be able to find her room now. Volume. We need to find that I am the one concept under cross section or how you got your the eye of the jungle Eggs half the base. Time to hide under basic question Heard. So what? You need to square off the the land on the green line here. So I need a four one minus next square. So the bottom. Now we&#x27;re going to the integration. It was from menace 1 to 1, and then the unsightly concluded, too. One witness, X square date. Next. So we will have a gun too. Jew X minus. Next. About a change of every tree there. That&#x27;s from minus 1 to 1. So we&#x27;ll have a thing called Eun Ju. No sign. One minus one by one with tree. My ass minus one. That&#x27;s one month, but three. Well, so I should get going too. One. My house. My house. One will be you on minus one, minus another one. We missed you, but three. So we should get equal to four minus for on the dream would be today on times. Do me drown Terminus for right cross fire

Taylor Shimono · Answer

So for this problem, we want to begin by drawing a sketch of Archie graphs. So we know that wise equal to X squared is just going to be a problem. Um, and it&#x27;s going to be centered around the origin with the Vertex of the origin. And it&#x27;s going to look about like this. And our next equation is the graph wise, equal to four, which we know is just going to be a horizontal line right here and so we can see that the intersection of our points is going to occur from X is equal to negative to two X is equal to positive too. And we know that we want to find the volume that&#x27;s going to result by building. I saw sleaze triangles so we can sort of picture like this. I&#x27;m on top of our area between these two graphs, and we know that the formula for the area of these ice Ossan strangles is that our area is going to be equal to the have pot news squared and then divide that by four. And so to find the volume of this region, we&#x27;re going to essentially just integrate the area over the ex Interval. So the integral from negative to positive too, of our iPod news, which we know is just this upper line, which is why is equal to four minus or lower graph, which is why is equal to export. So just four minus x squared squared over four DX. And so to interrupt this, we just want Thio distribute out the numerator. And so this is going to become the integral from negative to to to of 16 month eight x squared, uh, my herpes B plus x, 1/4 power all divided by four DX. And when we finish integrating this, we&#x27;re just using our power rule. We&#x27;re going to end up finding that this is going to be equal to for X minus 2/3 execute plus one over 20 x to the fifth power, and this is all evaluated from negative to positive, too. And so evaluating using our fundamental during practice, this is going to end up being equal to eight of minus 16 over three, plus 32 over 20. And then it&#x27;s going to end up being a minus eight uh, plus 16 over three minus 32 over 20. And this is going to end of simplifying Thio tell us that our volume ever sold is equal to 128 over 15.

Di Zhang · Answer

Hello, everyone in this very I&#x27;m going to show house off problem eight. In this question, it is known that their reception perpendicular to X axis is the ecological time go to accomplish is Ariel. We need to find out the lance off the side of this equilateral triangle writes in the Unit Circle. We draw a right triangle nowadays, the one next. It&#x27;s just the distance from the original denoted backs. The other act, you know, T It&#x27;s easily we can express end in terms of X Richard Square root one minus square. We don&#x27;t have that lance off the site of the The curse section is double. Then afterwards, we can booked out there the area of the core section, which is the scrub three over to Air Square. Next, we&#x27;re going to characterize the modem integrating the area of the cross section, which can be express being this four. Okay, it&#x27;s not difficult. Oh, Cutler. The intro, which is four over three screwed threat

Taylor Shimono · Answer

So for this problem, we are going to want to begin by making a sketch of the set up of our problem. Because we know that our base is formed by the semicircle wise equal to the square root of 16 minus expert I&#x27;m And so that&#x27;s just the top half of a circle that&#x27;s going to go from X is equal to negative for two. X is equal to positive for um and it&#x27;s also bound by our X axis. And our last piece of information that we&#x27;re given is that our cross sectional area perpendicular to our X axis. So that means going this way. That is going to be a bunch of squares. So essentially where you are building squares on top of this shape here and we know that the base of each square is going to be this distance right here, which is actually just equal to our FX. And that means that the area of one of these squares, because of the fact that it&#x27;s a square, is just going to be f of X squared. And we know that in order to find the volume of a shape, we&#x27;re going to find the integral of our area. So that means that in order to find the volume of the shape form by building a bunch of squares on top of our A semicircle is going to be the integral over our X pounds, which go from negative 4 to 4 of F of X squared and so F of X is a square root of 16 one&#x27;s expert. So when we swear that it&#x27;s just going to end up being a 16 minus X squared the ex And so when we integrate, this is going to be 16 x minus 1/3 X cube evaluated from negative four to positive. For so using the fundamental theorem of calculus. This is 16 times for which is 64 minus 64 over three Martus. 16 times negative four will be negative 64 um plus 64 over three, which is going to end up reducing to a total of 256 over three as

Gregory Higby · Answer

okay. I think the toughest thing for this people for these problems is is realizing that you have two different functions that you need to do. Ah, the semi circle that&#x27;s on top in a semi circle. That&#x27;s on bottom. Um, what&#x27;s nice, though? Is the formless pretty easy? Ah, that you&#x27;re going from Negative on the one, um, And then you have to take your top function minus your bottom function. Um, and it&#x27;s in terms of X because it&#x27;s perpendicular to the x axis. Um, and what you&#x27;re really doing here is you&#x27;re finding the length of a side length and you&#x27;re finding each each individual square and then the the integral adds them all together for you. Um, so what&#x27;s the upper function? Well, if you sell for why, in this problem, you would first want toe subtract x squared over, but then you would square root to sell for it. Now the positive one will be the upper one. And why equals negative square root? Cause don&#x27;t forget when you square root, you get to answer is positive and negative. So the positive, the above semicircle, the negative gonna be the below semicircle. So it&#x27;s nice about this. Um, and you might just need to go to a calculator. Um, is your upper function of one minus, uh, X squared, And then you subtract off the lower function negative square root of one minus X, where there&#x27;s really the same thing just right over here cause it&#x27;s starting to get busy. It is adding, because it will you subtract the name you add, so there&#x27;s really from negative 1 to 1 two of the square root functions that need to be squared so you can cancel out that square root with this as long as you remember to square this and then distribute that four in or really looking at Interval from 81 toe one of the term four minus for X squared is two. Squared is 44 times. One is 44 times X squared is four X squared DX, and then we can do this math for X minus 4/3 X cute, and that&#x27;s from negative. Want a one when you plug in 14 times one is four minus 4/3 because one cubed is still one, but then you need to subtract off. Negative four becomes plus 4/3 So what do we have? Will four minus 4/3 if you get the same denominators of me, 12 3rd for this for is the same as 12 3rd Minus four would be a third&#x27;s, and then this will become minus negative. Same math, just negatives. A 38 3rd plus 1/3 is 16 3rd which you can already tell. I had certain I had written down. Just double check my work before I started getting into it. Um, so how do we get to part B here? Well, the only thing that&#x27;s different, all the work is the same. All the work is the same. The difference is the formula for a semi circles pi over eight in front. Everything else is the same. I actually just got puts. Quigley&#x27;s in here top minus the bottom. That being squared. All of my work is the same. If your teacher makes you do all the same work well you have to do is take 16 3rd Your answer got earlier times. Pi ovary. There&#x27;s a reason behind it. I don&#x27;t have the time to get into that. Um, but 16 acres and 16 2 times. And look at that. You get to pi or three. The answer. I already wrote down

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.

Problem

The base is the triangle enclosed by $x+y=1,$ the…

Problem 9 Medium Difficulty

Find the volume of the solid with the given base and cross sections.
The base is the unit circle $x^{2}+y^{2}=1,$ and the cross sections perpendicular to the $x$-axis are triangles whose height and base are equal.

Answer

$\frac{8}{3}$

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