find-the-coordinates-of-the-centroids-of-the-given-figures-each-region-is-covered-by-a-thin-flat--13

Name: find the coordinates of the centroids of the given figures each region is covered by a thin flat 13
Uploaded: 2019-08-15
Duration: 2 min 39 s
Channel: Numerade
Description: Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region bounded by $y=2-2 x, x=0,$ and $y=0$ about the $y$ -axis.

Question

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region bounded by $y=2-2 x, x=0,$ and $y=0$ about the $y$ -axis.

Amy Jiang · Accepted Answer

I guess you were asked a surface area. It&#x27;s a integral of We&#x27;re getting from zero to the square roots of two to buy time to our function. So that&#x27;s explored. I am the square root of one plus part option trying to x and it didn&#x27;t have to Do we get for export the ex? Yeah, that&#x27;s what two X equals intentions of you. Then we have that to the ex physical to seek it squared you Do you? I can&#x27;t say we get the integral Yeah, tangents squared you pi over four spirit of one plus tangent squared you secrets with you you as we need changer and points He had that at exit Todo tangents of you is also equal to so So we have a musical about extradicted discovered up to we have you is equal to tangent inverse of too screwed up to get no putting it in our own points We had champions never said she was going to And Joe, this is equal to just I over four integral of sequins. Why you wanted to get you? Do you do you that ticking in a row of this we get This is pi over four times one over four seasons. She did up engines of they Don&#x27;t minus one over h si Jin&#x27;s there. Tangents of data minus one over AIDS. Ellen Secrets. You know, those tensions data? All about what? It attendant. Hi. Invested too high, Richards. We&#x27;re rid of two. And before this, infinitely. Thank you. Why don&#x27;t you get that? Approximately 13 0.99?

Bobby Barnes · Answer

we want to find the volume of the solid generated by all being the region. Why is even X squared the X axis and ecstasy to around the why access. So this here is a sketch of that region and the first single gonna want to do is right out what equation we want to use. So since we&#x27;re rotating around the why access here, this chapter says we should use the equation four interval from a to B of some outer radius, which I&#x27;ll call Capital R. And we want this with respect. Why minus some Inter Radius Joe called Little are and I will also be with respect why? And we also need to multiply this by pi. So the first thing we&#x27;re going to need to do is figure out what our radius is are going to be. So we&#x27;re going to start from the live access and then just draw a little lying to the first thing we hit. So the first thing we hit is why is even two X squared? So one thing we&#x27;re going to need to do with this is rewrite it so we can get it in terms of the function were actually interested in. So we want to rewrite this in terms of why So I&#x27;ll go ahead and square root each side so end up with screwed of why is equal to X. So the distance from the ex access to this blue line should be the square root of why. So we can go ahead and write that our little why should be the square root of why and we&#x27;ll do the same thing to figure out what our outer radius should be. And it just ends up being X is equal to. So are capital R is just going to be two now we can go in and blow these values and so high I&#x27;ll leave all our bounds and integration is gonna be for right now. So our outer radius was too. So squaring that gives us or are in a radius should be the square root of y and squaring that will just give us why. And then we have our do I on the end here. So, looking at our graph, what we need to do is figure out what our range of why values will be so in this case, one of them is fairly obvious. It should just be Why is equal to zero since remember, we&#x27;re also bound below by the ex Access, and then to find our upper bound will need to find where the Line X is equal to two. And why is he going to X squared Inter sex each other? And so all we need to do for this is take two and plug it in. So I&#x27;m gonna take that to plug it into their and that just tells me why is equal to four. So our upper bound for this will be four. Now we can go ahead and integrate so high, so integrating four would give us or why and then integrating. Why will these power rule? So now it&#x27;s going to be, why squared and then we divide by the new power. And then we want to evaluate this from 0 to 4. So I&#x27;ll go ahead and plug foreign. So four times four is 16 minus four square to 16 divided by two will be eight. Then we subtract off when we plug it zero. And so for why zero and zero squared zero. So the second term just become zero and the 16 minus eight is eight. So we end up with a pie. So the volume of this solid when we revolving around the why access will be a pie.

Robert Leedy · Answer

Alright. So first of all, we&#x27;re given a poor abdullah or give in our function X squared like was X squared. Rebound by white was zero and X equals two. So we have our X squared function but we&#x27;re bound by X equals two and why equals zero. So we know this distance here is to when we rotate about the X axis that helps us know our radius. But we also know that up here we should be at y equals four. So are vertical radius. Here we take our cross sections is going to be four our maximum degrees All right, but our radius is given to us by the function X squared. So why equals X squared now setting this initial problem of pretty straightforward We&#x27;re bound from 0 to 4 and it is going to be pi times X squared dx pretty straightforward integral mover pie out front from 0 to 4. No, that&#x27;s squared DX. We get pie times X cubed over three from 0 to 4. You type this into a calculator, so we get four cubed divided by three. And that is approximately 64/3. So our solution part A is 64 pi over three. Mrs. Party, part B, we&#x27;re gonna do in white in part B. We are rotating about the line, or why equals four. So we&#x27;re actually rotating about this line right here. And no, if you&#x27;re not actually gonna have a hold so we don&#x27;t have to worry about that inner radius. We just need to worry about this outside radius so you can and go ahead and set that up and we want to worry about it can. But it should equal each other because there is no actual difference. So we&#x27;re gonna dio our function. Why? Minus four? Because that was our line when we get Y minus four. So we have X squared minus four. Far outer radius in foreigner. It is the distance from our ex access to our line, which is simply four units. So now we&#x27;re gonna set this up. I&#x27;m gonna leave the pie out front just to save us a little bit of time. Our radius is going from 0 to 4 of X squared, minus four minus from zero for four. All right, Now, neither of these intervals are too difficult to calculate. I&#x27;m gonna find him really quickly. But your fastest option is honestly just using your calculator. So we&#x27;re gonna get pie Times X cubed over three minus four x evaluated from 04 minus four x evaluated from 0 to 4. They were mostly everything by pie. So I&#x27;m going to type these into a calculator really quickly. Give me one second to complete that. All right. Typing that into a calculator, we get 34 0.13 repeating and we have this almost played by pie, which we&#x27;re going Convert that into a fraction and write the Pyin. It&#x27;s a curve inverting that into a fraction we get 512 pi over 15. Let me

Bobby Barnes · Answer

So we want to find the volume of the solid generated by revolving the region about the line. Why is he going to the square root of two? And the area that we&#x27;re going to rotate about this line is going to be bounded above by why is he going to the square root of two and below by why is it good to seek it? Tangent X. So the graf I have on the board right here is what we will be rotating or is the graph of our function. So the first thing that we should do is identify what formula we want to use in this case. We&#x27;re going to want to use the integral from A to B Oh, hi. And actually let me write the pie on the other side of the introversion or outside of edition, say, and then some radius function with respect X squared. So the area that we want to rotate is this inside area right here and we want to rotate it around. Why isn&#x27;t to the square root of two So what we want to do is start from the line. Why&#x27;s it to square two and go to our next lot. So this here is going to be our radius. And that radius would just be equal to the square root of two minus the bottom bound, seeking ex tangent X. So we know what are radius should be Now what we need to do is find our bounds of integration. So we know we want to start from X is equal to zero, and we&#x27;re going to want to find what the X value here is. So already have it labeled as high fourth. But the way I decided I want to do pie. Fourth is I went ahead and set seek int ex tangent X equal to the square with two and you can go about solving this however you want. And from when I went about solving it, I had just found that X is equal to high. So I won&#x27;t for you with the algebra there. But we just end up with a pie for at least for this first point in the first quarter. All right, so now let&#x27;s go ahead and plug everything in. So we have pie and we want to integrate from zero to pie. Our our it&#x27;s going to be the square root of two minus seeking ex tangent X the ex and I want to square this whole thing here. Now, let&#x27;s go ahead and swear that really quickly. And in doing that will get seek it square Tanja squared X minus two times the square root of two seeking ex Tendran X And then lastly plus so not the square root of two. Just two x. So what I&#x27;m gonna do is integrate each of these individually, and then we can go ahead and plug everything back in. So I&#x27;ll go ahead and call this one This too. And this here the read. So for integral one. And I&#x27;m just gonna forget about the bounds of integration right now. So I just want to find out what is the definite Inderal of seeking ex 100 square Dex E X. Well, you might notice that if we let you equal to Tan Drin X, we could do a u substitution and or our differential Landau with D. U. Is equal to seek it squared x d x. And then we can rewrite this integral here as the integral Oh, so I&#x27;m gonna replace you, Square her. I&#x27;m gonna replace tenant with you. So we end up with you square and then the sequence squared. X and the DX is what I have on the right hand side here. So then we could just throw on our d&#x27;you and now integrating. That is a lot simpler since it&#x27;s just going to be the Powerball for integration. So you now to the third power and then we divide by its new power. And since it&#x27;s a definite interval, be Adeline are plus C. But well, we&#x27;ve plug it back and we won&#x27;t need that part. Let me go ahead and actually pull this up on another screen to do the next one. So for the actually, let me go ahead on coffee, not over. It&#x27;s well, just so we could kind of keep track of it because it&#x27;s not wanting to coffee over. So object. Pretend my I think so, too. We want to solve for the integral, uh, seeking ex tender X yucks. And luckily for us, this is actually a known anti derivative. And this is just seek it sex, pussy, and then or our last part Mistress check. That was again. So it was two. And so then the integral of two. The ex. Well, that should just be two x plus seat. So now let&#x27;s go ahead in right everything out. So we should have hi times our pi times the interval of zero to let me just make sure what the bounds for again pie for Oh, So, actually, one thing I forgot to say was on this part we don&#x27;t care about you squared the care about what? You is so tattered. So I need to go ahead and take this and plug it back in. So that ends up with 1/3 Tanja Cube X plus C. And so that&#x27;s what I want to write it in next part. So it&#x27;ll be 1/3. Kendrick cute looks. And since we&#x27;re doing I&#x27;m sorry. We don&#x27;t want to write the integration anymore. We already integrated everything. So here, one right, 1/3 Candra Cube looks and remember, since we&#x27;re now working with, this is a definite interval. We don&#x27;t need right plus C, since we&#x27;ll just cancel out. Wouldn&#x27;t be so sure. Then we should have minus two square roots of two. Seek it. Thanks, plus two X and we want to evaluate that from zero to pie. So you still have the pie out here now Pie for of attention of pie forthis one so well, first, just end it with 1/3 there seek it is one over co Sani. So just go ahead and write that so I don&#x27;t get on overcoat side of X, and so pie for of that will be the square root of two over too. But you can go ahead and write that as so minus two Sperber, too. Times two over the square root of two plus two times pi Or all right, that&#x27;s going to be the first part. And then we need to subtract off when my plug it zero for everything. So zero, um tangent pretended of zero will be zero. So I just have to zero in the first spot and then minus tooth where would too so co sign of zero is one so won over coast side or one over one is just one. So we just left with too much screw you and then plugging in zero or two x will just have zero. Now we can just go ahead and simplify something so the pies should still be out there will still have that 1/3 Linus. So the square root of to hear the square to their counsel out and then two times two is just or and then probably four times to should give us plus pi over, too. And then we&#x27;re gonna subtract. Well, in this case, when I distribute those negatives council out. So we just added, we get to square groups of two. And so now we can just go ahead and combined these constants right here to get Hi. They&#x27;ll be negative. So 12 11 over three waas. Hi. How plus two square groups of two. And you could do some algebra to rewrite this however you want. But I&#x27;m just gonna go ahead and leave it like this here. And that should be our volume when we rotate about that region

Robert Leedy · Answer

all right. The function were given this time is why I equals the square root of X. And our bounds are going to be Life was two and X equals zero. Why equals two and X equals zero. So drawing this r squared of x function looks something like this. So we&#x27;re bound by X equals zero, which is our y axis. And why equals two which that will mean that we get for out here. Alright, now part is rotating around the y axis. What you&#x27;re gonna do this similarly, Teoh, the X axis accept our radius is gonna are bounds for Integral are gonna be our radius here from 0 to 4 rather than on the y axis. So we&#x27;re gonna go from 04 are function is gonna be pi times the radius squared are radius is given by this function. So squared ax squared DX. Now that&#x27;s actually making her life very simple because you&#x27;re gonna cancel each other out. Always simplify if you can, so we&#x27;re gonna pour pie out 04 and we have the square root of X. So that is just going to give us X dx. All right, so when we already value and we have pie times X squared over two from 0 to 4. Pulling in 44 square to 16. Divide that by two. We get eight and then plug in zero. It&#x27;s all zero. So we end up just getting ate pie for our volume of part A For part B, it&#x27;s a little bit different because we are bound by a different line. We&#x27;re bound by X equals four nerve mounted at pound were rotating around X equals four. So actually rotating now here, which makes life actually not too bad. Um, so we&#x27;re still going to find our outer, which is gonna be four minus r radius. So four minus r radius, which is the function here, and that is gonna be our outer radius. And our inner radius is going to be the distance from our Why access to our line, which is just our, which is our inner radius. Now we&#x27;re going to go ahead and set up our interval problem, which gives us from 0 to 4 are outside pi times four minus the square root of X squared DX minus are integral 04 high times four squared DX type This into a calculator. So again grabbed a graphing calculator on minors. Math nine song and hit math. Nine. Now, I&#x27;m actually going to not type in the pies because that will make our calculator and not give us our answer. I&#x27;m gonna pull that out first. Pull it out and we get I&#x27;m sure before four minus spirit of acts squared DX minus 04 for squared DX. All right, type this into a calculator. So 04 or I&#x27;m gonna four minus the square root of X squared. Make sure you put in parentheses, so we square all of it. DX minus math nine 0 to 4 again or squared. DX ID equals tacular is taking a little bit toe. Figure this out and I get my solution is 34 0.6 repeating pie, which we can convert into a fraction 34.6 repeating man. Fraction me gets 104 pi over three

Problem

Find the coordinates of the centroids of the give…

Problem 24 Easy Difficulty

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region bounded by $y=2-2 x, x=0,$ and $y=0$ about the $y$ -axis.

Answer

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