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1. Find the x and y coordinates of the center of mass of theregion R bounded by y= x and y= x^2 if the density of the region is(x, y) =12x^2y^2....

Question

1. Find the x and y coordinates of the center of mass of theregion R bounded by y= x and y= x^2 if the density of the region is(x, y) =12x^2y^2.

1. Find the x and y coordinates of the center of mass of the region R bounded by y= x and y= x^2 if the density of the region is (x, y) =12x^2y^2 .



Answers

Find the mass and center of mass of the lamina that occupies the region and has the given density function .
$D$ is bounded by $y=x^{2}$ and $y=x+2 ; \rho(x, y)=k x$

To begin their problem. Let's write, Don't hypothesis. We have. We have the easy Andy, which is bounded by Why is it called to one minor success Quiet, which is super abuela? And why is it called zero on the density it's given by Roll off X comma y going to Kate and Sway? So if I draw the picture so the region very interested in So this is my wise ical to one minus excess square and we're interested in this street. So the intersection off the X axis and the parable. So let's find out the intersection Point Select said them equal to each other. One of our exist questions come to zero the same place, except I called a plus minus one. So this point corresponds to negative one on this point corresponds to positive for so next step. We weren't going to find the most of this region. The moss in is a culture integration from negative into positive one. And so that's my ex. Limit the wine images from 0 to 1, minus X Esquire Key times Y Do you like DX? Let's take the key outside. It's a negative 12 positive one. The integration of why is simply why it's quite divided by two 10 0 to 1 minor success. Square T X and this becomes Q divided by two negative. Want a positive one? One line a success Square Hole Square, DX. The next thing to observe here that one minus exist quite is even function. So the inter function is even function, So this becomes two times K, divided by two 0 to 1 one minus six. Escort, hold skirt DX and to win two cancels. Okay, so let's write down the formula for one one way in a sexy squad. Hold square. It's just given by one plus extra depart for minus two ex Esquire DX. So if I intricate them, this becomes X classics to the par five, divided by five minus two. Excuse, divided by three and the limiters from 0 to 1. If I substitute upper limit minus lower limit, then it becomes one plus one of her Fife minus two over three. But it's simply a tiki divided by 15 said last night. What? Let's find out. Ex cording it off the thinker off muss so the X coordinate is given by one over him. So the mosque Integration from negative 12 Positive one 10 0 to 1 Minor success. Square Qiwei Times six. Do You I followed by t X So then let's bring the key outside. Secure for him Negative onto positive one X types Life squared, divided by two religious from 0 to 1 minor success square and followed by ttx Okay, it's a cure for him now the integrationist some negative into positive one. This becomes x two one minus exist square on the whole square, divided by two DX observation here is one minus X Discredit save in function. So they're saving function. So they changed the color. So their stay even function. This one is the art function. So this in place this inter function is out function. So then on dhe, if we integrate your function in a symmetric interval like negative into a positive one. So this become zero. So my ex co ordinate is equal to zero. So let's find the white cord in it next. So I called in it. It's going to be won over him into question negative onto positive one 0 to 1 minus x Esquire Key Wife Square. No. Do you I d x Because, remember, So the white cord in it becomes 10 city times y Sochi times, white times white. So that's why I have this extra y squared salutes. Evaluate the integral secure for him as the trolling in outside. Negative on the positive one. So this becomes why Cube divided by three leader to one minus 60 square T X. So what is that? This becomes Q divided by three times in negative into positive one one man has access card Whole cube DX. Let me go back and check you. Stop our limit one minus six escort and then hold Cube. And I can always pull this three outside, so there's exactly I'm doing. So no, this becomes unless you start property that there Steven function so too. Okay. Divided by 3 a.m. So then 0 to 1, one minus. If I don't the formula here, this becomes three Xs cleared. Plus three extra depart for minus six to the boar, six DX. Okay, so, no, let's intricate them. This becomes X minus X cube plus three over five extra depart five minus extra deep or seven divided by seven and from 0 to 1 on defy, tow the integration and put up I limit finest store limit. This becomes simply 16 divided. Play 35. Okay, so let's put the values for I am so more precisely one over him. So tok, divided by 3 16 divided by 15. 35. Be justice diverted by 35 times. One over him is going to be 15 divided by eight key. I can cancel the key. I can cancel the four. I can cancel them. Forceful. So then this Fife and seven. So this becomes finally for over seven. So that's my white coat in it. So my ex coordinated cereal on my Y coordinate is for over seven. So that's just enough, moss.


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