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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.
The region bounded by $\quad y=\cos (2 x)$ $x=-\frac{\pi}{4}, \quad$ and $x=\frac{\pi}{4}$

Vikash Ranjan · Accepted Answer

since no density Funston is given. So we will assume that the reason has a uniform. Dante, that is density is the constant not from the figure. This is the figure we can see. The boundaries off X exists. The morning is is minus by a four by four way See that the reason has symmetry with respect due y axis This symmetry so like this. So this is the symmetry. This line divides the two part of the graph this rentable order. So this is the symmetry to Texas now on DDE hence ex body questioned you. Therefore we needed to calculate Why won&#x27;t one Lee since we is you? That the reason hat uniform density Therefore come on, My bar equals one upon toe into you&#x27;ll off. It will be affects. Sorry, Whole Esquire dicks Ban here, Toby. Offensive deice equal This when about to integer off. Minus pirate fool to find four causes Squad Rex digs a bomb you do love minus by by four you buy my food because two ex index So this is eagle do 112 integral off minus by. Wherefore by a pawn forward, huh? Cause for X plus one dicks a band didn&#x27;t love minus two by four to find by four cost to extend ears. Sins cost to Tita. Equus. Oh, sorry. Since cost of tea date was to costs for Tito minus one. Therefore, we have costs Party diapers cost duty dark plus one upon you. So we have this thing Aspirin, this formula. No, this is equal. Do when a born fool you jiggle off minus primal four by one Food. One less call for this DX about Sign to ex upon to minus pile one for toe by apart. Now this is equal do one of 14 x plus sign for ex upon four minus by upon for by a bond forth upon when I want to signed to X minus Bible for to buy a pawn four on this is equal Do one of one before. Why? Upon food last one upon food, A sign off according to buy a pawn for minus one upon four into minus pi upon for plus one upon four. Sign off for in tow, minus by upon forth. Bam! When I want to do in tow, Sign off. Doing two by four minus Sign off. Doing minus by 14 Eso This is equal to one of one Ford in tow by upon four plus Jiro minus one upon forward into minus pi upon for plus judo. When I want doing toe one minus off minus one. And so this is equal. Do fire upon 16 blessed by upon 16 upon. One. This is equal do to buy upon 16 equals by upon it. Therefore year center of mass As it&#x27;s about Inouye Vay, this is equal to do and by a point Sansa 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.

Mauricio Araiza Canizales · Answer

in the following question. We want to find the center of mass located in the intersection between the functions. Why? Sickle to cosign effects on bicycle to native Kocian effects for X in between negative pirate chewed to pirate tube. We&#x27;re getting that The density in between this reed in between these two functions is equal to two. So it&#x27;s first pro on a proxy in a sketch of her intersection. Are the region are, which is the intersection of these dysfunctions? Looks like So we&#x27;re here. We&#x27;ll have the point. Negative one y school too negative one and one and then point pie Sequel to two Hi, Divided by two sorry and negative piety Right about you. So we know that cosign effects starts from one and then a pirate to zero. Then it repeats itself and it&#x27;s the same around the negative. That was Fricks. So this is the line. Why sickle to co sign effects with green. And when I draw the line, why sickle to native coast and effects, which is the same. However, it&#x27;s just reflected across the Y axis, so this will be the graph of negative ghost and effects. So this is why sickle to negative coast and effects. And this will be the region. Are Did you notice something similar in our well are has symmetry. Sorry around the X axis. And why does bar has symmetry around the X axis? Well, we know that this shape right here is the same once this one And because of that, we can see that it has similar around the XX is however eat old soon has symmetry around the y axis. And we know that because this shape right here and this one are the same. So because our has symmetry sorry, because this two functions have symmetry around the x and Y axes. We can actually compute the center of mass without actually using any integral. And in this case, is gonna be the intersection of this juice lines of symmetry which in this case is just one on this one, and they intersect at the 10.0 comma cereal. So because our because these two functions have symmetry around the x and Y axes, the center of mass is located at the intersection of these two alliance which

Amy Jiang · Answer

Okay, so you&#x27;re wise. You could sell X is equal to zero and x squared over four plus y squared over 91 yet So we&#x27;re only looking from this region, this region. So we&#x27;re only looking at Cochin one so x squared over four that gives me a radius of two in a radius of three. We have 1/4 of a circle and where has to find the center of mass of that. Okay, so let&#x27;s be writes this equation in terms of why so we get just why is equal to three square root of one minus X squared over four. Okay, now let&#x27;s find export and white boy, so we have export. That&#x27;s gonna equal in a row from our intervals. So that&#x27;s from zero zoo to of x times our function. So that&#x27;s three x square root of one minus X squared over four X and also in a row from 0 to 302 Just a function which is three square root of one minus. X squared over for the ex agrees his cancel. So we get this is equal to three times negative for over three times one minus X squared over four to power three over to all over. Actually might be gone. So another X over too. Brian Scar Bit of one minus X squared over four. But sine inverse of x over to these are all the different to todo. Okay. And the simplifies to just h over three pie. Sounds fine. Why, boy, that&#x27;s 1/2 times integral from jokes. Two of our function square jaw. That&#x27;s 911 is X squared over four all over three. They go from job to this can cancel. Just turned into three to get three. Open to Okay, so yet that times in a row of three or nothing just involve scar Widow one minus x squared over four. The ex get integrating us. We gets X minus, X cubed over 12 3 over to all over. It&#x27;s high over to Okay, And please evaluate this from two to dope. So we get this is six over three pi over to which becomes four over pipe. So are senator of ashes. Eight over three pie home before over pine

Vikash Ranjan · Answer

Listen, we have the opportunities excess squat upon fool. Unless what? It&#x27;s what upon mine equals one. This is Eagle Do. Why squat upon nine? He was one minus. It&#x27;s just got a point, fuller. Or why waas so realize what it was nine minus 914 eggs, Esquire and what it was three drove off. Sorry, this is three and do, and it&#x27;s going to be have one minus exit squad of particles. No, this is the graph. Why was three I&#x27;m just quadrant one minus exist choir. Eventful. So we have this rough, uh, tree and do a wound one man. It&#x27;s it&#x27;s quality before. No, since no. Then two different Sam is men sent. Therefore, we assume that the density is constant. Dad, we wouldn&#x27;t find Well, bye. No finding white body we have. Everybody was when one door and take that Toby if X homeless while dicks God and did it in to be and 66. But this is equal. When upon to India, Finest going to do and a record tree. Do one minute, 60 squad apartment food dears. My indeed you&#x27;re minus student tool. Three little one. Man. I think it&#x27;s swell before the X. So after father&#x27;s well being is indeed. And we obtained this as nine by two. Interior minus 2 to 2. One minus X Esquire 14 dears. All right, bye. Three. And the dealer that his minister to do. And the route one minus its plaid on board dicks. So this is equal Do doing nine by two. You didn&#x27;t do because we are changing the limits, sorted the property off. It is a giggle that whenever the Hansen is even then we can gender limit and we can we&#x27;ll go by the bowling deal by too. So we arranged this here. This is Stan in this problem in the three in the little the little group and then one minus its is God on four years. So for the solving, we obtain this as had, just cancels out, says six. So we have front this as nine under bracket XY minus 114 in tow. Excuse me. Three. Let me dizzy, though, to do on upon we have six on the record X might do until bread and a good one minus its esquire upon fool lesson sounding worse except one too. And you don&#x27;t don&#x27;t go for authorities. None to do minus Q upon three upon six in Digital. Plus, I want to kiss his. He could do do it upon three by this becomes for upon by. Therefore we have center of mass, as do you do in for the fight dancer really given

Mauricio Araiza Canizales · Answer

in the following problem, we went to find this interview mass of the region Between the grafts off bicycles Zira Excess sickle deserve an X squared divided by four plus y squared, divided by night sculpture one. So let&#x27;s first start off by making a sketch of this graph. So we know that this crafty sexually Ellen the lips with its major access a line in the wind direction. So we know the one of experts is located at zero comma 30 common negative three and the other two around two comma zero native to calm a serum. So let&#x27;s join this points. Now we know the line. Weiss equal to zero is just the X axis. Just this one right here. And we know that the line waste equal to zero is just the You&#x27;re just the XX, sis. Sorry. The ex The line exit ical diseases the y axis, which is just one right here, and the region in between. This three graphs. It&#x27;s this one right here. This one. Sorry, just took one. So let&#x27;s first determine a function for this section right here, so we can rewrite. This is why circle too, that why squared divided by nine plus X squared divided by four is equal to one. No, let&#x27;s abstract excess square divided by four remote sites will have one minus X squared, divided by Ford. Then we&#x27;re gonna multiply everything through by nine. So have widespread support to nine minus nine x square, But it by Ford. And finally, we&#x27;re gonna take the square root of both sides. So have nine minus nine divided by four x square. And this is also equal to three over two times the square root of four minus sexist where so in this case, we&#x27;re gonna take the positive sign for this a square root which is going to represent this line Over here. It&#x27;s another. We have this information we can actually start by finding the finding the center of mass. So you know that the export, innit of the center of mass is equal to the moment. With respect to the Y axis, he read about a total mess of the system. And the white co ordinate of this insurance mess is the moment with respect to the X axis divided by the total mass of the system. So let&#x27;s just find him so and musical to the integral from A to B of room times FX minus D effects the X In this case, this is gonna be our FX energy effects is gonna be all our lower function. So G affects the function below this one. It&#x27;s gonna be the line Why is equal to zero and because that is zero, then we can just cancel this term. So we&#x27;ll just have the integral from Syria to two of Roe camps three over two times the square with the four minus 60 square the ex This integral is actually difficult to compute by hand a serve Ragno. So basing a computer software program or graphing calculators you can find out that this interval is just three ex champs for minus X square plus strove rx sign Aw, thanks. Divided by true and all of this has been divided by four We&#x27;re going to Bali. These from super to truth. If we substitute these values will find out that M is just what 23 pi divided like true Temps Road. This is our M value. So now let&#x27;s find the moment. The Frighteners Sorry, The moment would respect to the y axis this quantity is given by the integral from a TV of Roe Times Road temps X chimes FX minus. She affects again jfx zero. So we can just get rid of this tube right away. There&#x27;s gonna be the integral from 0 to 2 of room jams three halfs thames extend to square with the four minus X elsewhere. The ex In this case, we can actually use your substitution and see that you is gonna be for a minus six square So the u D X is gonna be native to X. Therefore, the U it&#x27;s gonna be negative. Two ex tempts the X, so the year is gonna be the red opinionated to notice equal to x, the X. So after the substitution, we can rewrite. This is the integral of Roe terms. Three halfs times native. 1/2 came to square with the few do you Right now I&#x27;m not going to changed back to change the the limits of integration. The boundaries. Because of the end, we&#x27;re gonna go back to X. So let me just finish this part of the problem in their widlitz. So the integral of this term&#x27;s just negative. Three forts attempts room terms. 2/3 tempts you to three caps, which we can turn back into her ex. This terms just gonna be negative. One has of road times four minus x elsewhere Racing three halfs evaluated from your room to true, which just end up being for Rome. Finally, we need to find just the moment with respect to the X axis. In this case, this is gonna be 1/2 times the interval from a TV of room. Tim&#x27;s f affects a square minus ye of Texas, where yikes again, these terms just zero. So this is just going to be the integral one or two from 0 to 2 of room chimes. Three halves swear rid before minus six square Swear the X. This simplifies to nine over eight champs the integral from certitude of road times four minus Texas where the ex this is gonna be nine. Number 8 10 through times There terms the integral. Sorry if I evaluate this integral, we&#x27;ll have four x minus X square. Sorry. Excuse me. Waited by three ballet the from certitude by using power room in my simple find. This term right here, we&#x27;ll end up with nine over eight Times row champ. 16 over three, which just turns out to be six row. So now that we have determined this three values can actually subjective them and find the X and Y quoting its prisons are a mess, So we&#x27;ll have four row the right of way three pie, bro. My victory coma six row divided by three pie row Do you like? True? So the ex and white co ordinates for our center of mess We&#x27;re just gonna be eight divided by three fine coma four divided by pi This is a solution to the question.

Problem

For the following exercises, use a calculator to …

Problem 273 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.
The region bounded by $\quad y=\cos (2 x)$ $x=-\frac{\pi}{4}, \quad$ and $x=\frac{\pi}{4}$

Answer

$\left(0, \frac{\pi}{8}\right)$

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