for-the-following-exercises-use-a-calculator-to-draw-the-region-then-compute-the-center-of-mass-o-11

Question

For the following exercises, use a calculator to draw the region, then compute the center of mass $(\overline{x}, \overline{y})$ . Use symmetry to help locate the center of mass whenever possible.
$[{T}]$ Region between $y=\sqrt{x}, \quad y=\ln (x), \quad x=1$ and $x=4$

Mauricio Araiza Canizales · Accepted Answer

in the following problem, we went to find the center of mass of the regions in between the grafts off bicycle, just quarter defects and wise people to liberate natural logarithms off Alex for ex imaging white wanting for. So let&#x27;s first start off by making a sketch of the graph. So we know that why sickle to excess square looks something like this unless actually pinpoint the point one coma want. No, we don&#x27;t. We know that l natural algorithm effects is zero with one and that it continues to grow. So we&#x27;re here. We have the 0.0 Sorry. One comma, sirrah. And over here we have the intersection at zero comma zero. So we&#x27;re looking at this point right here. Now, I&#x27;m just establish that this is a point x equal to fort, so we want to refine center of mass of this region right here. Now we have the German that we need to find this interference with its white with x X and y coordinate. And we know that it&#x27;s X coordinate. It&#x27;s gonna be equal to the moment. With respect to the XX is sorry to the Y axis divided by the total must&#x27;ve the system and why it&#x27;s gonna be equal to the moment would respect to the X axis divided by the total mass of the system. So let&#x27;s first thing the total mass of the system and which is equal to the integral from a TV of the density of the quake off the function right here Times If X minus G affects, Yanks in our case will have em equal to the into girl from 1 to 4 off row. The density trumps a square with defects minus ln effects. Yes, the first integral is easy to compute, so we&#x27;ll have row times. The integral over square root effects is 2/3 temps extra three halfs if we use Powerball and now the integral of negative natural logarithms effects by using a graphing calculator or computer software program to find it is made of X times. Alan Effects was X. None of this is being evaluated from 1 to 4. If the approximate values will end up with them equal to 2.1 to 1 row. Now I&#x27;m going to use another whiteboard. You calculate the moment with respect to the white necks this physical two days to the Inter integral from a TV of rope times X temps ffx minus G affects the ex. Now let&#x27;s have sifted. The balance overhears We&#x27;ll have the integral from 1 to 4 of row temps X Times Square with defects minus Ellen effects The ex The moment would respect to the Y axis can be written as the integral from 1 to 4 of Roe Times X to the three cafs minus X temps. Ellen effects the ex The first, integral again can be computed easily. So this is gonna be equal to two over five terms. Ex rest of the 5th 5 over too. And again, we need to use a graphing calculator computer somewhere program. Do you actually find out this integral which is equal to negative one over true excess where Ellen Effects plus one over four excess where all of this is being evaluated from 1 to 4. If we substitute, these values will end up with the moment with respect to the Y axis equal to 5.6 times rope. Now let&#x27;s find the moment would respect to the X axis? This quantity is equal to 1/2 attempts to integral from from a TV of rope times F f X square minus gene affects the square Yanks. If we substitute, we&#x27;ll have Dane tickle from 1 to 4 of road times the square of this world. The fix is just X and the square natural logarithms just ellen off excess Where the ex now this integral the 1st 1 just becomes square. It affects Sorry, excess worried, Divided by two. The second, Integral will again need a use of a graphing calculator, a computer somewhere to find it. And it is equal to X terms. Ellen affects a square plus two X temps Ellen effects minus do X, and this is being evaluated from 1 to 4. And if you plug in all of its values, you&#x27;ll find out that the moment would respect to the X axis is equal to 2.452 Temps Road. Now that we have determined this Valley&#x27;s, we can actually start plugging in to find the coordinates for the moment, for the center from s. Sorry. So the 1st 20 is gonna be 5.6 Thames row derided by 2.1 truth. Sorry to 0.1. Shoot one rope coma, 2.452 temps around, which is just this quantity over here, divided by 2.1 to 1 roll. So the coordinates the excrement of disinterred messages. 2.3 years. 86 after simple fighting, this fraction of mercury and the other one is 1.156 And this one right here is a solution to this question.

Amy Jiang · Answer

Okay, so we&#x27;re asked to job region. So we have wise equal to the square root of X and Y is equal to island of X, and this is from 1 to 4. So let&#x27;s see that this is from one before we&#x27;re gonna have something. See, I believe our reboot of expansion would feel like Bess Intern are Ellen. Function would go like this. So we&#x27;re looking for this region over here, right? This is a skirt of X, and this is Ellen of X. Okay, So since there&#x27;s no cemeteries were gonna just calculate our center of mass directly So expert is equal to the interval from what did we say? It was once before, uh, x times our top minus our bottom graph. So that&#x27;s extensive Root of X, which is extremely over too. Minus x ellen of X e x over Inter Girl from just X 1/2 minus Ellen of x The X once before. So this gives me to over five x 25 over to minus one. Have X squared Ellen of X minus one over four x squared all over to over three extra three over to punish ex Elena X minus X all about what it evaluated up from four. It&#x27;s one, yet it simplifies to see the Tickle to 3 to 3 over 20 minus eight Owen before over 47 over six. It&#x27;s what it&#x27;s for. Ellen of four. Okay, so that&#x27;s approximately equal to 2.2. And now we have y bar that&#x27;s equal to 1/2 in your little phone once before. Over top function Square two This is X minus R bottom function, which is owned of X Squared the X all over in a row from what? Four of X 1/2 oneness. Elena vets. Okay, what does that give me that? Give me 1/2 times one. Have X squared minus ex Ellen squared X money to excel in X plus two X All over 47 are actually that&#x27;s 47 over +61 is for Elena Fort. We don&#x27;t need to evaluate this from 4 to 1. Okay, this becomes 1/2 3 over 21 is for Ellen, where four plus eight. Evelyn of four over 47. Over six minutes. Four. Ellen before which is approximately 1.7 So our center of rash becomes 2.2 comma. 1.7

Amy Jiang · Answer

Okay, so we have wise equal to two X and two X squared Y is equal to zero. And let&#x27;s see, explica todo to one. Okay. Were asked to draw out the region and in computer center of Mess, and we could use somebody to help locate the region. Okay, so let&#x27;s draw out our graph first. Well, two X squared is a problem. We have something like this, and then it cuts up between Joe and one. So it&#x27;s returned. One is over here. So we&#x27;re looking at this region over here. Okay, Well, looking at this, it seems like we don&#x27;t have any cemetery, so because we don&#x27;t have any cemetery we had to use, um, any properties or any shortcuts of calculating it, So we&#x27;re gonna do it directly. So we&#x27;re gonna take export to equal interval from Gerald to one of X times. Why, that&#x27;s two x cubed over the integral of just Joe&#x27;s one from of exports eggs. The X does that Give me that. Gives me. Mmm mmm. Oh, it is it to over here to actually Cassidy Stew&#x27;s. Okay, So I had just the interval from Joe&#x27;s one of Excuse that you explore over four over X three over three, evaluated at one, and it&#x27;s usually one over four over one over three, which is equal to three over four. Now let&#x27;s find White Boy. It&#x27;s equal to 1/2 of the interval, just one of two X squared times X ray just to be ex. All right, which is execute the X over in ago. Joe&#x27;s one of two X squared DX. What does that give me? That gives way. Oh, wait, no, I&#x27;m squaring it right, I&#x27;m staked. So I get actually to exploit the power of Jews. That&#x27;s four x for X over Indigo from Joe&#x27;s one. Just why would just two x squared x Okay, so we get well, our 1/2 can&#x27;t so that this because it too. And then I choose chances that would get X over five over five extra para fireball in five over X to part three over three about looting at one and tell excuse me. Three over five So our center mass becomes really over four comma, three over five

Mauricio Araiza Canizales · Answer

in the following problem. We want to find the center of mass of the region in between the functions. Why is equal to two X square and waste equal to zero for X in between zero and one. So let&#x27;s first draw sketch for this graph. So over here we&#x27;re going to draw the functional bicycle to Texas Square, which we know is the problem and the journal for myself. Gravel. It&#x27;s gonna be this one right here. And let&#x27;s just a point to substitute X just to pinpoint. So we&#x27;ll have the point X is equal to zero in the point X equal to one. So let&#x27;s find what is the white co ordinate for this point. So the Y sickles zero. We substitute that over here. Well, find when zero. I&#x27;m gonna use Rhett to determine at this point. And now at X equal to one y sickle to two. So we&#x27;re here with a 0.2. Call me one. Sorry. One Coma too. The 0.0 commence here. Now the line y sickle to zero is just the X axis. Is this one right here? So the region aren&#x27;t that we&#x27;re trying to find its center of mess is this one right here? So we have drawn the function in the graph. Let&#x27;s find out the center of mess. So we know that the center of mass it&#x27;s located. But this coordinates and we know that this special cases of X and Y are located at the moment with respect to the Y Axis awaited. But the total mass of the system and the way coordinate it&#x27;s located at the moment would respect to the ex access be raided by the mass of the system? So let&#x27;s now calculate these three quantities we&#x27;re here. So the start of by calculating the moment would respect to the wax. This is equal to the integral from A to B of the density of the density of the function. Tim&#x27;s X temps, the upper function Buying is lower function times DX, in this case are bounce. We&#x27;re gonna being from Syria to one Temps road Temps X. So FX is gonna be to excess square And gee, if X is gonna be zero, So we&#x27;re just gonna leave that a serious now we have the integral from Syria to one of Roe terms to excuse the ex we can now perform this integral, which is released to compute, which is just gonna be extra fourth divided by four which in this case of was ballots are invalid from Syria to one. And if we succeed that these will have to roam times one over four, which is just roll over too. In their white words went to calculate the mass of the system. Just gonna be the integral from a TV of room tempts the upper function minus the lower one. The bounce are going to stay the same So from Syria to one times, Rowe says We know that ji, if X is equal to zero, would just substitute over here to exit square the ex. We can now perform this integral which is just X cube divided by three invalid from Syria to one. So the mass of the system is just your row terms one be better by three, which is just 2/3 of Rome. Finally, let&#x27;s calculate the moment with respect to the X axis. Now this is equal to 1/2 times the integral from a TV of roe Times Square fffx minus the square of GI effects the ex We can now sit sit through their bounce. And since we know that GI effects zero, we&#x27;ll just have to excess square square. This is equal to 1/2 Tim&#x27;s room times for extra the fourth. Yes, sorry. Forgotten DX over here. So the moment would respect to the X axis. It&#x27;s not just went too weak to Thames Row times this integral right here, which is just X to the fifth power divided by five validated from Syria to one which is just to row. He read it by five. Now we have calculated all of these three quantities. We can find out the location of the center of mess, which is going to be We know that X according it&#x27;s gonna be at the moment of inertia. Would respect to the Y axis derided by M and why is gonna be the moment of inertia with respect to the X axis? You bet it by m. So the first term is gonna be a road divided by two divided by to Rome, divided by three the white coordinate It&#x27;s going to be to row invited by five divided by two row, divided by three. The center of mass is just then we&#x27;ll have three of Rome divided by four row coma. You wrote temps three debated by five, divided by to row. So in this case, the X coordinate is gonna be three divided by four. The white coordinates gonna be three divided by five. And this is a solution to this.

Vikash Ranjan · Answer

we can see that the reason has a cement truck retrospective y exes. So the center of mass, you know, expert wipe on must be on the white sees and hence explain it was due. Therefore, we need to go plate one. A white hot. No. In this case, my bar is given us. But upon toe, integral A to B the fix squad. My nest. Two years squad dears. Ah, home indecision. Off effects minus Jax Index. So this becomes why he was upon to integral minus to do fire. Squire minus find upon for X squad. What a squad. The ex it is next, my ass to do five minutes fire upon four eggs. Esquire, dears. The gums. My body was when I&#x27;m gone too. Addition, Minus Toto, too. 25 miners identified by 16 extra power forward dicks. Bone minus studio doing Diggle. Five minutes. Five upon four extra Squire. Yes. So, first of all, we will see whether this council is even cancer or not. If if X is a even fansand on into a monastery, then integral minus into a FX DX egos tries off Joe do a FX DX. So this is the property off different individual. So we will apply this property in this above in Diggle I could get diligent. Therefore, it becomes two into 112 on off radiation were banned this as 25 x minus 25 bond extending into its to the bar five on five. So he had doing too. A bomb drives into five eggs minus fight upon foreign to excuse you upon three. You know don&#x27;t do so after putting this values will get the result which he&#x27;s like. This disease 50 minus. Why? By 16 into 22 minus Tito upon 20 miners 10 by 12 into eight minus You know this is equal to 15 minus 10 upon 20 minus 20. Bon three, this is Eagle Do Foodie upon for Dean to tree. This is equal to three hence, the singer of math is given by expert and this is Eagle do Joanne three. This is the answer to the given problem

Vikash Ranjan · Answer

being at the graph like so And in the graph, this is the graph off X square and this is the truffle exp artful. So you have to find center of months so Constable will find ex bad, which is equal. Do indeed your one x d m a bomb indeed. A little one on the end. So this is equal doing todo one ex indu. If this world minus six powerful index account and didn&#x27;t you don&#x27;t know why Exits by monastic spot for mix So this becomes indeed I did have one X Q minus six Parr file. This is about and you don&#x27;t have one. It&#x27;s just fine. Minus six. Party for weeks Dr. Swarming this We get expert foot upon four minus explore six of 16 You know it a while. Excuse you want to read minus X or I want five years. So this becomes 114 minus one upon six gone When I want to read minus 115 This is equal. Do six minutes before a bomb printed 41 fine minus three upon 15. This is equal due to upon 24 in 2 15 upon tow. This cancer started becomes three. Engine becomes It is a difficult Oh, So I think this is fine. This is equal to five over it. So you walked in? Why? My people&#x27;s sonny? This is It&#x27;s buddy was fine. Appointment? No, we have dropped in. Why buy so for opening will ever be, uh, this as and did a little toe one y d e n Upon indeed a little one in this is equal to deal dear one. Why do y to the power of one of 14 minus one of bomb? What about you wanna do? Do you want a form? Indeed. A little one. What did the apartment up on? Four minutes right to the ground. When? Upon new deal. This is equal to indeed a 0 to 1. Why did the power 514 minus y did a power 312 Do I? You find one? You do one. What did the apartment up on four minutes. Where did about one upon you do it? No. Resolving this aggression we have this equals 4.9 white about 914 minus to work. One fight party. The power five upon to You don&#x27;t know one you want to buy forth upon. Fine. Why did about five upon food minus? Do it? Want to be one of the power three upon to you don&#x27;t know what this is equal to four upon nine minus two upon by Defy divide. What upon fine minus two upon three. This is equal do to upon 45 defined by do a 1 15 So this becomes weapon for defiant into 15 upon To discuss is artist becomes one of one today. Therefore we have center mass as expert and why Bob is equal to fire bonnet on one of want to be so this is an answer to a given problem.

Problem

For the following exercises, use a calculator to …

Problem 276 Hard Difficulty

For the following exercises, use a calculator to draw the region, then compute the center of mass $(\overline{x}, \overline{y})$ . Use symmetry to help locate the center of mass whenever possible.
$[{T}]$ Region between $y=\sqrt{x}, \quad y=\ln (x), \quad x=1$ and $x=4$

Answer

$(2.38,1.15)$

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$(2.38,1.15)$

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Integration Review - Intro

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Catherine R.

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Gilbert Strang,

Calculus