find-the-coordinates-of-the-centroids-of-the-given-figures-each-region-is-covered-by-a-thin-flat-p-3

Name: find the coordinates of the centroids of the given figures each region is covered by a thin flat p 3
Uploaded: 2020-06-17
Duration: 54 s
Channel: Numerade
Description: Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=x^{3}, x=2,$ and the $x$ -axis

Question

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=x^{3}, x=2,$ and the $x$ -axis

Ernest Castorena · Accepted Answer

for this question. They want us to compute the area of the region bounded by. Why is it quits? X Cube X is equal to three in the X axis. All of you know about this cube function that&#x27;s gonna keep increasing above zero. And that&#x27;s why they chose to cut off. Had a vertical line. Three. So to compute this, they give us above the X axis. So he just computer as the area of the integral from zero degree of X cubed DX. Well, unless he was the funds Months knows the arm of calculus. We&#x27;ll end up with anti derivative X for over four, evaluated at three and zero. But if we evaluate at zero, we&#x27;re just gonna get zero. The only endpoint we care about the evaluation of his three. It&#x27;s the 1st 1 to be positive, and we&#x27;ll have three to the fourth over. Four. Well, that&#x27;s just 81 over for awesome. That&#x27;s the area in the region specified

Joshua Wheelwright · Answer

Okay, so now we&#x27;re going to find the area of the region enclosed by the X axis and the function. Why equals X cubed minus X So immediately, I&#x27;m gonna factor this. We want to find the zero. So this is gonna be X times X squared minus one and go further with that&#x27;s That&#x27;s X times X minus one X plus one. So there&#x27;s a different squares there. Always. We have three zeros, right? Ex kitty Cool. 01 and negative one. So we gots. See, um I don&#x27;t know how to make this smaller. It&#x27;s gonna be huge. Oh, no, it&#x27;s not You. Okay, so I got one there. Say that zero. Here&#x27;s one. And here&#x27;s negative one. Okay, so it was their functional click X cubed minus X. Well, if we know our polynomial sze, we can see that the a value is positive and it&#x27;s on degree. So it&#x27;s going to go something like this. I want to place you there from a cubic So I&#x27;m looking for the area enclosed by this function and the x axis were really looking for the area in here and in here. So how do we do that? We&#x27;re gonna take the definite integral from negative 1 to 0. That&#x27;ll tell us the first area there and our function is X cubed minus x d x and how to get the airfield 01 We actually are going to subtract because it&#x27;s under the curve there. We want to make it positive, too. Come to measure the areas. We&#x27;re going to subtract the definite A girl from 0 to 1 of the same function. If they&#x27;re both above the X axis, we would be adding. But because this one is below, we have to foot it. Bye. Ah, foot. Sign there. So it&#x27;s subtracting. Okay, so the anti derby will use the fundamental theory. Um, that&#x27;s gonna equal to the antidote. Motive. Eso here, the anti derivative of X cubed minus X. That&#x27;s gonna be X to the fore minus x squared. Um, that should be over for Ashby over too. I&#x27;m gonna evaluate that. A negative one and zero. And then what did we get from that? We&#x27;re going to subtract here. We&#x27;re gonna have the same thing extra before. Over four, again, minus x squared. Thank you. Wow. Over too. And this one&#x27;s made from 0 to 1. So I just put all those numbers in. Now Sewell gets on the left side. They&#x27;re really a zero to the 4/4, minus 0 to 2 over, too. And then within. That&#x27;s we have two minus. So you&#x27;ve used water brackets here. Such is the first part of the first part that we&#x27;ll get negative one. Negative 10 boy. I need another rocket there to the fore over for minus negative. One squared over to. That&#x27;s the second part of the first part Minus. Now put big brackets here. Okay, first part, the second Bert gonna put in one. So one, 24 Over. Four, minus one to the two over too. And minus zero to the four over. Four. Minus 0 to 2 over too. Okay, so I run out of room here. I&#x27;m going to raise the factory work, I guess case or and move up there and was gonna simplify all of this now. So first part there is cruelly just 00 um, minus now. Got negative onto the four, which is paused of what&#x27;s okay. Minus. But this brackets No wonder the force positive ones. That becomes 1/4 and a good one. Squared is also positive one. So let me minus 1/2 chase. So all that is negative. Zero minus what&#x27;s in the brackets? There. Now we&#x27;re in. Subtract the big brackets, and and there we get 1/4 minus 1/2 Um, which is a negative. That&#x27;s a negative quarter. So negative. 1/4 and minus What&#x27;s in these brackets? That&#x27;s zero again. Okay, not so bad. So Ah, yeah. Look, I just said 1/4 minus 1/2 his negative quarters, so we get negative. Negative quarter minus negative. 1/4. Okay, So negative. Negative quarter is 1/4 minus. Negative quarter is another 1/4 on those together, you get 1/2 okay?

David Marsella · Answer

So here we have the region to find by the parabola X equals Y squared minus four and the line y equals X over three. We want to find the area of this region Eso first, we&#x27;re just gonna examine it, decide whether we&#x27;re gonna do this disrespect to extra why our equations haven&#x27;t been given to us in any specific order. Uh, but if we do, our vertical and horizontal Sinovel wind test will find that if we do this with respect to X, we will need to regions one here and one here. However, if we do it with respect toe, why we can perform it with a single region, Uh, and therefore only one integral. So this looks like initially this looks like this is going to be the simpler way to do it. So that&#x27;s how we&#x27;re gonna do it. Which means we did our line in terms of why so X equals three. Why? And we know that our general form, in terms of life, our area is gonna be the integral from a to B of f of why minus g of y do I. We have our two equations already. Uh, the on our horizontal line. The right end of our line touches X equals three y. So that&#x27;s gonna be our f of why and the left side, the most negative side of our line or end of our line. It&#x27;s our problem. So a parabola is gonna be our G avoid. Now we need to find our A and B, which are going to be these points here. So we&#x27;re gonna have to calculate those. So we set them eagle to each other. We have three y equals y squared minus four, which rearranges into y squared minus three. Why minus four equals zero, which factors into y plus one time is why minus four equals zero. So we have a why of negative one and four. So our boundary points for this are gonna be here at negative one and here head for so ever and B so we can fill in our integral. So we have the integral from negative one, 24 of three. Why minus why squared minus four de y. We can distribute that negative to get a simple polynomial, and we&#x27;ll end up with four plus three y minus y squared de y. This makes Thean a derivative, very straightforward. With the power rule we get for why plus three y squared over two minus. Why cubed over three from negative 1 to 4. Now we can perform our substitution ins. We end up with 16 plus 24 minus 64/3 minus negative for plus three halves, plus 1/3 after combining. All of these are area comes out to 125 over six and there it is. You&#x27;re done.

Stephen Hobbs · Answer

with a problem like this where one curve is to the right of the other one. In this case, X equals three. Why is to the right of X equals y squared minus for the whole time? And so we&#x27;ll set up our integral for our area in terms of why values and then doing so. The limits of integration will also be y values. So we&#x27;ll end up going from negative one to positive for of the right curve minus the left curve. And don&#x27;t forget to subtract the whole quantity. Why squared minus four? And this is all with respect. Why? The reason we did this is because if we were to do top minus bottom over here we have an issue because the curve is kind of on top of itself in that region. Tobel left. And so it can be a little bit of a nuisance. And so it&#x27;s better often in that case, that just go ahead and use the, uh, right minus left approach there. Okay, so from here, we could just take some anti derivatives and find out what our final result is. And this would give us an area of 1 25 six or the area between the curve that shaded and black. There

Stephen Hobbs · Answer

to find the area bounded between these two curves. We could first draftees and then notice that there&#x27;s two regions bounded regions that were interested in and in order to find their intersection points. Of course, they intersect at the origin that X zero, but we&#x27;ll go. We could always go ahead and set these two things equal to each other So we could say the re X minus X queued is equal to a negative X and then adding extra. Both sides, adding excuse to both sides will get four X is equal to X cubed and then dividing out one more X here and we&#x27;ll get X squared is equal to four. So now it&#x27;s a little bit more apparent that we&#x27;re talking about the intersection points at X is negative two and the X is positive too. So there is some symmetry here. So now that we have that, um, did you mean thing that we&#x27;re gonna focus on now is which curved is on top of the other one? Because to find the area under between Kurds, we could take the integral from negative 2 to 0 of the region to the left of the origin And if you&#x27;ll notice that the top function there is negative X and then subtract the bottom function, which is, uh, the whole quantity Don&#x27;t forget to distribute that negative to both three x minus x cute. All of that is with respect Toe X here. So this is be X and then Plus, we also have the area to the right of the origin from zero to to of notice that he&#x27;s just flipped on the top function is now three x minus X cubed. And then all of that subtract negative X, or you could just say plus X at that point. Okay, so really, I&#x27;m just kind of working through most of the set up for these one, since that&#x27;s the difficult thing. But as you&#x27;ll notice here, of course, you&#x27;re gonna combine like terms. So here we&#x27;re going to get a negative four x plus x cubed. And here we&#x27;ll get positive for X minus X cube and then anti derived and so on. And so the area under the curve In this situation, the total area for both of those regions will combine toe positive eight, although it will always be positive with area under the curve. So that is our final value

Problem

Find the coordinates of the centroids of the give…

Problem 14 Easy Difficulty

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=x^{3}, x=2,$ and the $x$ -axis

Answer

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