for-the-following-exercises-use-a-calculator-to-draw-the-region-enclosed-by-the-curve-find-the-area-

Question

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $(\overline{x}, \overline{y})$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.
Quarter-circle: $y=\sqrt{1-x^{2}}, \quad y=0, \quad$ and $x=0$

Vikash Ranjan · Accepted Answer

in the ghost chin sense no density. Funston is mentioned. Therefore will a Jundallah Chisholm has a uniform density two, that is density is constant. Now we&#x27;ll schedule the reason that is in this cage. It&#x27;s on here. This is a sketch operation. This is our reason. This is one. This is one. This is X equals judo. This is my class. Gino on it is like this. This is the center. Point are you can sit in the center of math lies over here. This is ex violent life Since why people do it is a line of symmetry. So the center of mass must be on this line on the building, too. Expand equals life. So if you will find anyone off these either expert or way But then you can find up. Pull off this inter amounts. No, We see that the reason is, Demetri, with respect Oh, y equals X. That is the center of mass equals expert and a wine bar. It must be under lying land. It is wine equals X. I was there in the white, both Islam and can see from the figure that this center of mass is on the line way. Physics? Yes. Texas 9.6 and the center of mass lies over hell. So we have. Why buy equals one. I wantto integer. It will be ethics. All is quiet, dears. Man it, Toby FX. Deeks, this is equal. Do one of one toe digital one a record on the route one minus six is quiet. What is RDX about? You don&#x27;t know why. What are one of my exes? Quiet, Ducks. This is equal. Do when? Upon to Indeed, you&#x27;re the one under record. One minus X squared. Digs a bone. This becomes X from the root one minus six Esquire. Blessed one upon. Do sign Waas. Thanks with just one. Now this is equal. Do one of gonna do X minus six? Secure on three years one upon. This becomes one upon two into one minus. So Judge Ito plus one of Gondo signing Worse one minus J lo. So this is equal. Do what I wanna do into one minus one upon to read, minus general upon by my fourth this is equal. Do when upon doing do took on three upon final four density will do 113 into four on minuses equal before on to the five. Therefore center of mass equals expert. Why are these evil dude for a country by on 4.3 bites? Therefore, we often this as the answer to wonder given problem.

Amy Jiang · Answer

Okay, so we have the falling functions. Y is equal to X. So we have something like this. And then why is he going to minus? Let&#x27;s get to minus X. So we have y es until bet Su inimical downwards. And we also have Why&#x27;s he could sell. So we&#x27;re looking over here. Get me redraw this correctly. Okay, we&#x27;re looking at this region over here. If you notice at the X is equal to y our exit equal to one the line. It&#x27;s the magic. So our son over a mess, but must be on this Fine. So we don&#x27;t export is gonna equal to one. So we just need to find a wine bar. So it&#x27;s solved about y bar is equal to 1/2 times the interval from always. Said So you wanted intersect. That&#x27;s from Phil to 10 wait, no. We&#x27;re gonna have to We&#x27;re gonna have two lines, because if you think about it or to inter both because this is the top for this portion. And then this is the top for this portion. That&#x27;s for General from there&#x27;s one of explored the ex general from 1 to 2 of two minus X acquired the ex all over in a row from Joe&#x27;s one of X t x just integral from 1 to 2 of to minus X the X What? Let&#x27;s give us this is X cubed over three, evaluated at one and don&#x27;t plus negative to minus X cubed over three, evaluated up to one and the bottom of the road is this one. So he gets, See, we just get right to multiply by 1/2. That&#x27;s 1/2 times one over three or so minus zero, minus one over three. This becomes this one over three. So our center mess is this one comma one over three.

Vikash Ranjan · Answer

no days, doctor, though. Simple symmetry about why exes. Therefore, what is that too? Quarter in and off? The singer of moms. It&#x27;s by Equus. You. This is the car for excess choir class y Squire. It was full. And this is the car for Esquire class. Quiet square. So we have to find the area. Well, this is it. We&#x27;re going to find the center of Mars. This Senate arisen. No find. Why am I on expert? So nobody goes. Do you know her too? Why Diem or what? You don&#x27;t do them. This is equal. Do and Dale didn&#x27;t do one. Sorry. No, this is equal. Indeed. A legitimate one under Beckett one minus. What&#x27;s got your power? When upon to into one my nest one minus y squared to the power. When upon do into why do you want a plus one to do for my asset? Why, Squire bar on upon do? Oh, dear. When did you down? Yeah, General One. Right for my house. Why squared to the power upon to minus won my last? Why? It&#x27;s part of the power when I want to do I class wanted to four minus Why? It&#x27;s part of a power upon. True. So this becomes, do you know to what minus? Why is quite a power when upon do why do I, minus 0 to 1 one minus y squared to the power when I want to. And the wind the wind, Right? Bye. Did you do one minus? Why? Spotted one upon the way minus. Do you have one one? Minus y is for the power when I want to do it. This is our in addition first. And this is our in division second. First of all, we will do the indication. First division fast sequence for my last. Why? It&#x27;s part of the bargain upon. Do you are for this last June. Why it was to sign Tita and the road becomes too cause Todt dark. Is it equal doing together four minus four Sense what Tito borrowing upon to do cause she definitely to be equals. Indeed, you&#x27;re doing too often. Tend to do course two detectives becomes dealer for for spotted a digital. No, it&#x27;s the guns full in Diga because it&#x27;s what he done did you down. And since we know that also today waas to cost Scott cleaner minus man Yeah. Cost wanted. I was caused Judita plus one of mom. This is equal to fall in. Indeed. Um, cause to treetop plus one a two. Did you understand? Equal. Due to digger. Well, bless past, don&#x27;t you down? Indeed. People could do The integration becomes t Douglas, and I&#x27;m gonna do is sign it won&#x27;t be tough. So this is equal do to endure you don&#x27;t bless So sign teda because Tito one dude, because I&#x27;m too today post Wasn&#x27;t it about it? That is to Kansas this to become said to into To Douglas Scientist, huh? And the boss Linda, What we have this equals to Indu sign Tita cost Peter has a Tita. So we&#x27;ll put the values off. Scientist and cost in terms of why this becomes drives into why you want to Under route one minus. Why squat upon four glass sign waas Why wait? This is equal to go into why? Upon to and there&#x27;s a route one minus. Why it&#x27;s got off one fool plus do into sign Waas while you&#x27;re born to this is equal. Do why under rude, full minus y Squires upon four plus two standing wears Why you want to. So what All this becomes when you go on to the dude full minus Why, Squire Plus to sign Waas. Why you want to? Similarly, we have I do this equal group until oh dear. To one well, minutes. Why is far delight? Well, a human late. Why? Because Santa for do it becomes costed out. Indeed, we have this us. I started this so number here. Yeah. Now this is equal to Indio When my nose sign squad t data the power. Gonna pawn to Indu here when I got to do is the power So this becomes costed out. Did he? So this is equal. Do under indeed. Do we have You know what one money sense but it becomes governor of the hospital Others after meeting that would be open to Constanta in because the this is equal doing legal costs squad eternity down and since cost 23 diapers to cost squad Tee dum minus one. So cause scar tea, darling believe will do cost to cheetah last one of our crew. So this is equal Do when I wantto deedle one bless costs due to down. Did you know this is equal to one upon toe Indu Tita bless When about to sign duty down It&#x27;s a sequel to one upon, To and design Tita cause to Douglas T Down Go santo Sign to the equals to Sonny Because Tito upon to between give us Senator it cost it off as a teacher. So after making this teachers into terms of wine we have half into why a little one minus y squared plus half something worse. And why are you? Anybody becomes minus one of on three into four minus y squad the power to do Ditto. Do mine as minus one upon three into one minus y Squire to the power three About to do one the wind by Why do under track it four minus? Why Esquire plus two signing waas. Why might do you don&#x27;t owe do mine. As have why one man is why you Squire Plus huh, son in worse. Why you don&#x27;t know why. So after putting all those limits the appendage What about as one of my three in tow? Eight minus one. About three, right by five minus by four. This is equal. Do several born treaty right by three. Bye bye. Fool, This is equal to 7.3 into for a bomb creep by. This is equally 28/9 by, so we often wind us equals. So the body cools 28 over. No, I Now we&#x27;ll move forward on we locked in expert study for a long time. The value off him which is given us. It was twice. And the record indeed. All due to do four minus one Squire to the barber off. How do I minus? Indeed? Oh dear. To one one minus y squad to the bar with one of my do. Do I. This is equal. Dude, do into Culebra sis record. Why do and root for minus y squared? Plus two signing wears. Why you want to? You don&#x27;t go to minus Why one minus y squared. Blast on a condo san in Worse Why you don&#x27;t do one? This is equal. Do twice you do by minus 514 This is equal Do doing toe 354 This is equal. Do three by upon Do units. What? No. We have the center of Mass as expert on wine bar. It was Joel T Indeed over nine by on we have m equals three. By what do unions squired? This is the also to the given problem

Amy Jiang · Answer

Okay, So we&#x27;re asked if iron this enjoyed of y squared plus x word secrets one. And also why squared X squared is equal to import. So we have two circles of radius to and radius one. So we have something like this and invidious what? You&#x27;ll notice that these two circles are centered at the origin. So they are symmetric recall supposed both the X and y axis. Because of us, we can conclude that the center of mass is equal to zero common girl.

Vikash Ranjan · Answer

and we have the graph and this girl phase off. This is off. Why does next? Andi, this is a wine, but it&#x27;s quiet. So we have to complete this in Europe. Mosley jizz expert on DDE So, first of all, a medical experiment is goingto dignity of one x d. M by need or want Indian. This is equal. So integral 0 to 1 X in connects minus six. Is squad nix a bomb? Indeed. I owe you one. It was minus 6 12 Dex, this is equal. Go. Indeed. You&#x27;re the one excess body minus X Q Dears, do you want X minus? X squad index. So this is equal to excuse home. Three miles expire for upon For here are the ones it s possible to minus X cube on three. You don&#x27;t want equal to when I&#x27;m gone. Three minus one upon for mine as your body. But I won&#x27;t do Minus one upon three minutes. You, This is equal. Do four minus three by do everything right by three minus 216 is equal. Don&#x27;t wanna come 12 into six upon one that becomes one upon you. Therefore it&#x27;s body personable. Now I have to complete my bar. This is my body becomes You&#x27;re the one idea you&#x27;re doing Indian. So this is just want to do You don&#x27;t know what? Why into would&#x27;ve wondered why Minus, why do I a bomb? Get one? Well, but why minus why delay? So if you could do, you&#x27;re the one. Why do the power three upon minus? Why squad do I about you? Don&#x27;t know? One well over half minus wide. What we have this equals to my fine divided a Barbara Fine one do minus like you want three. You don&#x27;t know what I&#x27;m buying. Do you mind? Three. While the power three my do my nest Why is quite a punch? So you going toe to my five minus one upon three minus zero. Buying to one train minus one upon toe. This is eagle Do six minus 51 15. Divided by four minus three by six. This is eagle one upon 15 into six. Everyone, this becomes do you find. Therefore we have the world&#x27;s injured mask. Expand wife I wouldn&#x27;t do when upon you and you were on fire on we have n equals. Did anyone explain? It thinks it&#x27;s quantity Yes, because it&#x27;s equal. Do excess farm. I do minus execute by three years. Disease one upon to money. Upon review, jizzy will do one of 16 you need to square. This is that s o to the given problem.

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

Problem

For the following exercises, use a calculator to …

Problem 285 Hard Difficulty

Answer

$M=\frac{\pi}{4}$ and $(\overline{x}, \overline{y})=\left(\frac{4}{3 \pi}, \frac{4}{3 \pi}\right)$

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$M=\frac{\pi}{4}$ and $(\overline{x}, \overline{y})=\left(\frac{4}{3 \pi}, \frac{4}{3 \pi}\right)$

Calculus

Grace H.

Anna Marie V.

Caleb E.

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Integration Review - Intro

Definite vs Indefinite

Gilbert StrangCalculus

Grace H.

Anna Marie V.

Caleb E.

Samuel H.

Gilbert Strang

Calculus