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2) Find che mass of and the center of mass of thin Plate of constant density 8 covering the given region y = 18 _ -3 4na } = ~22Mass...

Question

2) Find che mass of and the center of mass of thin Plate of constant density 8 covering the given region y = 18 _ -3 4na } = ~22Mass

2) Find che mass of and the center of mass of thin Plate of constant density 8 covering the given region y = 18 _ -3 4na } = ~22 Mass



Answers

In Exercises $18,$ find the center of mass of a thin plate of constant density $\delta$ covering the given region.
The region enclosed by the parabolas $y=x^{2}-3$ and $y=-2 x^{2}$

Okay. We've been fine. Asked to finally center of mass X bar y bar of a thin plate of constant density road covering a region bounded by the Y axis and the curve. Why, minus y cute Secret of eggs on the interval zero is less than or equal to. Why is less than or equal one. Okay, so I noticed that we're going from 0 to 1 and that the point 00 is on the graph. If I plug and y equals zero, I get X equals zero. And if I plug in, why equals one? I get X is equal to zero. Okay, so we're only interested in the part of the graft between zero and one. And if I plug in, why is equal to 1/2 sit right here? I get that X is equal to 3/8. So we're coming out here a little bit. So this is our then plate of cuts and density. Okay, so, um, first, we don't want to find the mass. So next we're going to find Cam. Okay, So recall that M is equal to row times wth e inter go from a to B of they top most function or in this case, the right most function minus the left. Most most function with respect to why. Okay, so this is the right most. Okay, so since we're doing integrating with respect of why we're going from left to right instead of bottom to top. Okay, So if we look back over here, we see that the right most function is why, minus y que And that left most function is just like it's just X equals zero. Okay, so you play this in. So this is the inner grow from zero toe one. Uh, why minus like you d y. And this will give us the mass of our plate. Okay, so this is row times when we integrate. Why would get 1/2? Why squared? No, we integrate. Why cute? Begin 1/4. Why did the fourth? We're evaluating that from 0 to 1, we plug in one, we get 1/2 minus 1/4. And then when plug in zero, we just get zero. So this is 1/2 minus 1/4 is just one force that this is road, but before and that's equal to Okay. Next, we need to find the moments Okay, so for we need to find m x N m y. Okay, so when we're doing it, uh, integrating with respect to why M X is the is equal to the density times the integral from A to B. Why? Times f of why do you want which is equal to row times the interval from 0 to 1. If we come back over here, remember, this is why I minus y cubed. So we're gonna multiply that. Why? Okay, so this is road times the integral from 0 to 1. Uh, why squared minus y to the fourth? No. Why? And this is equal to row times. The integral of y squared is 1/3 like you. The integral of what of the fourth is 1/5. Why did the fifth? We're valuing that from 0 to 1. Which is ro Times 1/3 minus one fit, Which is equal to to ro over 15. Okay, so am ex is equal to tea, right over 50. Okay, so next to find m y to recall that him, why, When we're integrating with respect, why is row over to times the integral from a to B of, uh of Why squared do you want? Okay, so this is row over to terms the integral from 0 to 1 of why minus y cubed squared. Okay, so first of all, we need to multiply that up. So this is a row over tea times the integral from 0 to 1 of Why did the sixth minus two. Why? To the fourth. Plus, why squared do you want? Okay, so now we'll go ahead and integrate. This is Rover Two. The integral of White Of the sixth is 17 Why? To the Senate, Minus two times the integral of y to the fourth, which is 1/5 word of 1/5 plus the integral of y squared, which is 1/3. Why cubed? And this is evaluated from 0 to 1. So when we plug in one, we get 1/7 minus two fists plus 1/3 which is equal to row over to times eight over one, over which is equal to four. Row over one over. And this is equal to m y. Okay, so lastly, we need to find explore. Why bar And remember that ex far is m y over him, and my bar is in Excellent. Okay, So from the previous page, we know that M Y is for room number one out of five. We know that m we look back over here is Rover for and then em ex is to row over 15 Divided about RO over four again. Okay, so all the Rose cancel that, obviously. And we're left with 16 over 105 and eight over 50. And this is our center of mass.

Okay, so we'd like to find the center of mass, which is X bar. Why? Bar of a thin plate of constant density that covers the region bounded by X equals Y squared minus one in the line. Why equals X? Okay, so first of all, we gotta figure out the point of intersections of these takers, so I'm gonna serve equal to one another. So why squared? Minus why is equal to y um We'll go ahead and subtract away from both sides. We have y squared minus two. Why is equal to zero or why? Times y minus two is equal to zero. So these two curves intersect when y is equal to zero. And when? Why is equality? And since they're both contained on the line, like was ex, we know that the actual points of intersection are 00 and two two. Okay, so we'll go ahead and Graff the curves now. Okay, so we have 00 and two two, and we know we have the line y equals x. And that we have. This is a parabola. It looks something like this. Okay, so this is our region right here. Okay, So the first thing we want to do is find Mm. Remember that M is equal to row terms the integral from A to B of f of X. We never Why, Sorry. Minus D of why d y? Okay, so in our case, this is ro times the integral why goes from 0 to 2. Okay. And then the right most function is X equals y and the left most function is y squared minus one faces. When we distribute the negative, we end up with ro times the interval from 0 to 2 of two. Why minus y squared do you want? Which is ro times. Now we're gonna go ahead and integrate the integral to Why is why squared And the integral of y squared is 1/3. Why keep? Okay, so we're gonna First of all, we're gonna play into we end up with four minus too. Cute is eight. Someone has 8/3 and we played in zero. We get zero. So this is equal to four row over three, and that's equal to our mass. Okay, so next we need to find the moments. Okay, So, um, let's see. That means M X when we're integrating with respect to why is equal to row times the integral from zero 22 of why times we look back over here times too wide minus y squared D y. So this is equal to row times the interval from 0 to 2 of two. Why squared minus y cute Do you want? Okay, so we'll go ahead and do this integration. So this is row times 2/3. Why? Cubed minus 1/4. Why? To the fourth evaluated from 0 to 2. So this is ro times you play gin to and why keep would get eight times. 2/3 is 16 3rd and then minus 16 force and we plug in zero. We just get syrup. So this is equal to four row over three and this is him. Thanks. Okay. Next we gotta find m y. Okay, so then why is equal to row over to terms the integral from a to B? Uh, f of why squared minus d. Why squared do you want? Okay, so we're gonna look back and see what all this is. All right. So we know that this is the interval from 0 to 2. Go back to our picture here um, hour, Right? Most function is why equals X. So that is why squared minus are left. Most function is why squared? Minus Why? This is why I squared. Minus Why Squared. Okay, so let's, uh, much blood that out. So we have Rover two times the interval from 0 to 2 of y squared minus. Boil this out. We end up with water. The fourth plus to why? Cute plus y squared do you want? Okay, so now we're gonna distribute the negative sign. So this is row over to times the integral from 0 to 2. The y squared. You're gonna cancel out and we're left with to Why? Cubed? Minus? Why did the fourth D why? Okay, so I will go into this integration. So this is row over to times we end up with two force. Why did the fourth we add one to the exponents divide by the exponents minus 1/5. Why did the fifth and we're evaluating that from 0 to 2. Okay, So noticed that to force is the same as one. Okay, so whenever we plug in to we get row over to times 16 over too, minus 32. Number five plug in zero would get zero. So this is equal to Rover two times eight over five. What? We can cancel here and say that this is for row over five and this is M Why? Lastly, we have to find the center of the mass, which is X bar. Why? Bar is equal to em. Why divided? I am comma M X. The What about him? Okay, so em Why we just found is for row over five and em. Our very first page is for row over three and M x is also for row over three. Okay, so when we invert, multiply, we end up with three this and what? So this is equal to explore what bar?

Okay, so we're asked to find the center of mass. The center of mass is X bar web Are a thin plate of constant density covering the region bounded by y equals X minus X squared. And why equals negative X. Okay, so first of all, we gotta figure out where these two things intersect. Okay, so, um, se x minus X squared is equal to negative X, and I'm gonna find the zeros. So, um, let's move. Everything over to the right hand side of zero is equal to X squared minus two x. So this is X times X minus two. And these two things intersect when X is equal to zero. And when y is equal to when X is equal to zero, why is also eat with zero when X is equal to why is equal to negative. Okay, so let's stroke the picture. Okay, so they intersect a 00 and it too negative to you. Now we know what the line looks like. Line y equals negative. It and the parabola. It is placing damn word and look something like this. Okay, so this is the first problem in the section that we had It's not symmetric with respect to the X or the Y axis. So we're gonna have to find both X bar and Weber. But the first thing we do is fine in on the mass to remember that the mass is equal to the constant density times the integral from a to B of half of X minus G epics D x. Okay, So in our case, um, the interval X goes from 0 to 2. So that's the integral from 0 to 2. Okay, so the top most function is the parabola, which is why equals X minus X squared. So that's gonna be X minus X squared minus D of X, which is native X DMX. Okay, so this is equal to row. Tells the Inter go from zero to, um, this is two x minus X squared. Okay, so when we integrate, we get row interval of two X's X squared in the interval of X squared is 1/3. Excuse me, and this is evaluated from 0 to 2. Okay, so this is row times score minus 8/3 in case of four is the same as 12 3rd Okay, so end up with him is equal to for row over three. Okay, so next we have to find the moments m x and M. Why? In case of M X is equal to road. I don't see in a girl from a to B of 1/2 times Affects quantity squared minus. Give IX quantities quick. Okay, so in our case, this is row over to if I pull up this 1/2 right here times the in a row from zero to let's look back F of X's X minus X squared minus negative X squared. Okay, so this is row over to times the integral from zero to, and now we have to foil this out. So we end up with X squared minus to execute plus X to the fourth minus x squared. In case I've noticed that the X squared cancel out right. And now we could integrate. So this is Rover two. The integral of negative to excuse is to do better before, which is 1/2 X of the fourth plus 1/5 exit with, and this is evaluated from 0 to 2. Okay, so we can plug in to and we have minus eight plus 32 over five. Okay, which is row over to times negative. Eight over five, which is negative. Four row over five. No, that's what we did here is just cancel wonder. So this is M X. Okay? We also have to find him. Why? Okay, So em Why is equal to row times the interval from A to B of x times f of X minus D effects D X. Okay, so this is Rome tells the integral from zero to of X. And you remember that f of X minus D of X is to x minus X squared. Okay. You can't just come back over here and check F of X minus D of X is equal to two x minus X clear. Okay, so let's distribute the ex. So this is row. That's the integral from 0 to 2 of two x squared minus X cube. Okay. And I will integrate with respect to X. So this is Rose times 2/3 X cubed minus 14 extra. The fourth and we're plugging in zero and two. Okay, so one of plague in two. We have rope times 16 3rd minus four. So this is 16 pirate Mona's 12 3rd or four. Row over three. Okay, so this is him. Why? Okay, so you'll recall that X bar is equal to em. Why? Over in why bar is equal to M X over it. Okay, so let's go back and see what all those things are. Okay, so em is four row over three him ISS for row over three. Mm x is negative for Rover five and em. Why don't we just found ISS for over three. Okay, so X bar is equal to four row over three. Divide about for over three, which is just equal to one and why war is equal to negative for row over five, divided by for row over three. Okay, so in Burton, multiply, this is negative for row over five times three over four road, which is negative. Three fists. So the center of mass export y bar is equal to one and negative three fists.

Okay. We'd like to find the center of Mass X bar. Why? Bar of a thin plate of constant density row that covers the region given by, uh, the intersection of these two curves y equals two X squared, minus four x. And why equals two X minus X squared. Okay, so the first thing we need to do is find the points of intersection. Then we need to do this so we can finally, uh, limits of integration for definite minerals. Okay, so we're gonna set to X squared minus four X equal to two X minus X squared, and we're gonna solve for X. So if we move everything over to the left hand side, we have three x squared minus six. X is equal to zero. We could factor out of three X and we have three x times X minus two is equal to zero. Okay, So that tells us that these intersect when X is equal to zero and when x is equal to Okay. So notice that if I plug X equals zero into either one of these equations say we'll plug it in here we get why is equal to zero. And if we plug in X equals two into the second equation we get for X minus for Or why is equal to zero again. So our points of intersection are 00 and 20 Okay, so we want to go ahead. Okay, So noticed that why equals two X squared minus four x is a parable of affected at a to you. I had X squared minus two x. And I want to complete the square because I want to figure out where the vertex of the probable ISS. So I'm gonna add one to make that a perfect square and noticed that what I really did is at two. So I have to subtract two to make sure I didn't change anything. So this is too times X minus one squared, minus two. And that tells us that the vertex of this parable A Is that one negative too. Okay, so we'll do a simple thing for the second curve, which is also a parabola. So here, I'm gonna factor out a negative of negative X squared minus two x and noticed that again. We need to add one inside the parentheses to complete the square. But what we're really doing is subtracting one. So we have to add one out here. Okay, so this is negative. X minus one squared. Close one. So this is a Perella that faces down with Vertex at one. Okay, so let's put all this information together too. Graph the region. Okay, So recall that the curves intersect at 00 in 20 though 00 and 20 Um, the first curve has the Vertex at one. Negative, too. Okay, so down here. So this is a parabola. It opens up, okay. And the next one has ever text at 11 And it is a parable of the open Stan. Okay, so this is our a rough sketch of our metal plate, so noticed that by symmetry is pretty easy to see. That ex far is equal to one. So what we have to do next is we have to find why are okay, so we'll do this in in steps. Okay, So the first thing we have to do is find the mass, so find him. Okay. So recall that mm is equal to the integral from a to B times are density function, which I'll call road times. f of X minus D of X d eggs where f of X is the top most want to. So this is the parabola that opens down. So this is why he is equal to two X minus X squared. Okay, so this is the interval from 0 to 2, and I pull row out front. And then we have two eggs minus x squared, minus the parable A to X squared, minus for X. Okay, so we're gonna combine like terms. Um, we do that. We're gonna have two X plus four x +06 X minus three X squared. Okay, so now we're in a great. So this is ro Times The integral of six X is three x squared, and you can check by taking the derivative. You do get six X minus. The derivative, the integral of three X squared, is excused. We're evaluating that from zero to t. Okay, so if we put this in, we get row times, we plug in to get 12 minus eight, we plug in zero would get zero minus zero, so we get that M is equal to four. Ready? Okay. So next we have to find the moment Mm X. Okay. So recall that M X is equal to the honor roll from a to B Thames row over to times f of X squared, minus G of X squared. Okay, so in this case, we'll pull Rover two out front, and this is a row over to tell us the integral from 0 to 2 of f of X is to x minus X squared, and then we're take square in that minus G of X is two x squared minus four X squared. Okay, so we need thio. Get ahead and foil that out and head like terms. Okay, So when you boil it out, we have a row over to the front, the integral from zero to. Okay, so we pulled this out. We'll have four x squared minus for excuse plus pics of the fourth minus will pull up the 2nd 1 We have four XO before minus 16. Excused. Plus 16 x squared. Okay, so we'll distribute the negative and gather the like terms. So this is row over to times the integral from 0 to 2 of we have four x squared, minus 16 x squared. So that's negative. 12 x squared And then we have negative for X cubed plus 16. So that's plus 12 execute. And then we have x of the fourth minus four x of the four. So that's minus three X to the fourth. Okay, so let's integrate. So this is ro over too. Times we integrate. Negative 12. Excuse. They have negative. 12 over three execute. We integrate 12 x skewed. We have 12 over four X to the fourth. No one to integrate. Negative. Three extra. The fourth. We have negative 3/5 Axel if it evaluated from 0 to 2. But notice that 12 over three is the same as four and 12. Over four is the same, s three. Okay, so we're gonna plug in our limits of integration, So we'll play again too. So end up with eight times for which is negative. 32. Plus, here we end up with three times 48 and then we end up with negative three fists. Times 32 which is negative. 96 5th We plug in zero into each of these would just get zero. So we have a row over to times negative 16. This or we have M X is equal to negative. Eight. Row over. Okay, so the last thing we have to do is find the center of mass. Remember that explore is won by symmetry. And why bar is M X over in, which is one times MX we just found is negative. Eight row over five and em is for So this is one and negative too. This is the center of mass.


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