5

Point) A lamina occupies the pant. of the disk 2? + y? < 16 in the third quadrant: Find its mass if the density at any point is 6.9 times the point's distan...

Question

Point) A lamina occupies the pant. of the disk 2? + y? < 16 in the third quadrant: Find its mass if the density at any point is 6.9 times the point's distance squared from the origin:Mass

point) A lamina occupies the pant. of the disk 2? + y? < 16 in the third quadrant: Find its mass if the density at any point is 6.9 times the point's distance squared from the origin: Mass



Answers

A lamina occupies the part of the disk $ x^2 + y^2 \le 1 $ in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Okay, So for this problem were given the equation X squared plus y squared is less than or equal to one. So since typically this ends up being are What we can say is that we know are is going to be in between zero and one. And what we also know is that data because it's in the first quadrant. So since it's going to be from just in the first quadrant, we're going from zero Tau pi over two for theta. Okay, So in order to be able Thio was asking us to do is to find the mass. So the double integral and I want to take care of our first. So it's gonna be pi over 2 to 0, and then 1 to 0 is I'm going to dio kr sign of data times are d r d theta. So the very first thing I wanna do I'm gonna pull sign out. I mean, I'm sorry. Pull k out and then I'm going to combine everything, so I've got to ours. So it's gonna be r squared sine data and then d r d theta. So I've got okay and then I want to go ahead and do the integral of our. So I'm gonna have one third are to the third and then sign data d r D data. So now I'm gonna pull this one third out as well, and I'm actually going to evaluate lips instead of d r. We wanna put from 0 to 1. So what I want to do is put this one third K and then I want to go ahead and value weight. Um, I want to go ahead and evaluate. This are to the third, which I know. Putting that in from 1 to 0 was going to be one minus zero. So it's gonna be one time sign, which is just sign. So this is going to end up being one third K And then, of course, the integral of sine is negative. Cosign data from pi over 2 to 0. So I know that co sign of pi over two is going to be zero, so I'm gonna have one third k so zero minus and then co sign of zero is one. And it's negative co science. So it's going to be negative one, So it's going to be positive one. So which means that my mass is going to be K over three or one third K. So now I want to go ahead and find the, um the ex portion of the center of mass. So I'm going to do my one over M or my three over K, my aunt a girl from zero to pi over two and then from 0 to 1. And I'm going to do our Kassian data because cosine is the ex portion and then kr signed Data times are D R d theta. And of course, instantly I have this K that I can eliminate many. So I have three from pi over 2 to 0 and then I've got 123 are. So it's going to be part of the third cosign data signed Data de are de Seda. Okay, then what I want to dio is I want to go ahead and take the integral of art to the third Power. So it's gonna be 1/4 art of the fourth Cosine theta signed data from 1 to 0 and then d theta so like I did before, I want to go ahead and move out this 3/4 and again this are is going to end up being one minus zero, which is one So it's going toe technically just kind of disappear. But I've left with co sign sign. But I know I have an identity for to co sign data signed data that I can use. However, if I'm gonna multiply this by two, I have to divide by two. Someone had to divide the 3/4 by by eight or by two. So now that's gonna become 3/8 instead of 3/4. Now, what I can do is I can rewrite this. So now I can rewrite this as a sign of tooth data D theta. So now the integral of this is going to end up being, ah, negative co sign of tooth data and then times one half and then from pi over 2 to 0. So I'll be left with 3/8. And then, of course, two times pi over to his pie. Co sign of pie is negative one. So this is going to be one hat positive one half and then minus so co sign of zero is also positive one. So it's gonna be times negative one half. Some have This is a double double negative becomes a double positive. So 3/8 times, Um, 1/4. I'm sorry. I'm sorry. 3/8 times. One half plus one Half is one. So the answer is going to be 3/8. Okay, so now we want to do a very similar thing to find the why portion of the center of mass. So I'm going to set up my one over the Mass, which is three over k and then my integral roles. And then this time I want to multiply by our sign data. And then my function KR signed data. And then, um, times are and then d r d theta do so if I noticed right away my case cancel and I have again three are so art of the third and then sine squared theta d r d data. So I'm gonna go ahead, integrate my arm, so I'm gonna have 1/4 art of the fourth. And who Sorry about that. And then sine squared theta from 1 to 0 d theta. Yeah. So then I'm also now again going to move out my 1/4 that's now going to become 3/4 and then I have this sine squared data. Now, before I do that, I'm actually going Thio. Um, go ahead. Let's switch this over. So what I'm gonna have is one half times one minus cosine. Two theta de seda. Yes. And I'm gonna end up moving out this one half, so I just want to integrate. So this is going to become 3/8. And so now when I go ahead, integrate this. So this is going to be theta minus, and then a coastline integrated is going to be sign, and then I have to integrate the two. So it's gonna be one half from pi over 2 to 0. So if I look at this, I have 3/8 and then I have data, which is pi over two minus two times pi Over two is pie. Sign it. Pie happens to be positive one. So I'm gonna have minus one, okay? And then we want to do minus zero minus sign of 00 So minus one half Mhm. So I'm gonna have 3/8 times pi over two. Yeah, and then I'm sorry. This So sign a two times pi over two is gonna be pi sign of pie is one. So this is gonna be minus one half. So then, if I look at this, I have minus a half and plus a half. So those cancel. So I'm left with 3/8 times Pi over two, which is going to be three pi over 16 Yeah.

Okay, So for this problem were given the equation X squared plus y squared is less than or equal to one. So since typically this ends up being are What we can say is that we know are is going to be in between zero and one. And what we also know is that data because it's in the first quadrant. So since it's going to be from just in the first quadrant, we're going from zero Tau pi over two for theta. Okay, So in order to be able Thio was asking us to do is to find the mass. So the double integral and I want to take care of our first. So it's gonna be pi over 2 to 0, and then 1 to 0 is I'm going to dio kr sign of data times are d r d theta. So the very first thing I wanna do I'm gonna pull sign out. I mean, I'm sorry. Pull k out and then I'm going to combine everything, so I've got to ours. So it's gonna be r squared sine data and then d r d theta. So I've got okay and then I want to go ahead and do the integral of our. So I'm gonna have one third are to the third and then sign data d r D data. So now I'm gonna pull this one third out as well, and I'm actually going to evaluate lips instead of d r. We wanna put from 0 to 1. So what I want to do is put this one third K and then I want to go ahead and value weight. Um, I want to go ahead and evaluate. This are to the third, which I know. Putting that in from 1 to 0 was going to be one minus zero. So it's gonna be one time sign, which is just sign. So this is going to end up being one third K And then, of course, the integral of sine is negative. Cosign data from pi over 2 to 0. So I know that co sign of pi over two is going to be zero, so I'm gonna have one third k so zero minus and then co sign of zero is one. And it's negative co science. So it's going to be negative one, So it's going to be positive one. So which means that my mass is going to be K over three or one third K. So now I want to go ahead and find the, um the ex portion of the center of mass. So I'm going to do my one over M or my three over K, my aunt a girl from zero to pi over two and then from 0 to 1. And I'm going to do our Kassian data because cosine is the ex portion and then kr signed Data times are D R d theta. And of course, instantly I have this K that I can eliminate many. So I have three from pi over 2 to 0 and then I've got 123 are. So it's going to be part of the third cosign data signed Data de are de Seda. Okay, then what I want to dio is I want to go ahead and take the integral of art to the third Power. So it's gonna be 1/4 art of the fourth Cosine theta signed data from 1 to 0 and then d theta so like I did before, I want to go ahead and move out this 3/4 and again this are is going to end up being one minus zero, which is one So it's going toe technically just kind of disappear. But I've left with co sign sign. But I know I have an identity for to co sign data signed data that I can use. However, if I'm gonna multiply this by two, I have to divide by two. Someone had to divide the 3/4 by by eight or by two. So now that's gonna become 3/8 instead of 3/4. Now, what I can do is I can rewrite this. So now I can rewrite this as a sign of tooth data D theta. So now the integral of this is going to end up being, ah, negative co sign of tooth data and then times one half and then from pi over 2 to 0. So I'll be left with 3/8. And then, of course, two times pi over to his pie. Co sign of pie is negative one. So this is going to be one hat positive one half and then minus so co sign of zero is also positive one. So it's gonna be times negative one half. Some have This is a double double negative becomes a double positive. So 3/8 times, Um, 1/4. I'm sorry. I'm sorry. 3/8 times. One half plus one Half is one. So the answer is going to be 3/8. Okay, so now we want to do a very similar thing to find the why portion of the center of mass. So I'm going to set up my one over the Mass, which is three over k and then my integral roles. And then this time I want to multiply by our sign data. And then my function KR signed data. And then, um, times are and then d r d theta do so if I noticed right away my case cancel and I have again three are so art of the third and then sine squared theta d r d data. So I'm gonna go ahead, integrate my arm, so I'm gonna have 1/4 art of the fourth. And who Sorry about that. And then sine squared theta from 1 to 0 d theta. Yeah. So then I'm also now again going to move out my 1/4 that's now going to become 3/4 and then I have this sine squared data. Now, before I do that, I'm actually going Thio. Um, go ahead. Let's switch this over. So what I'm gonna have is one half times one minus cosine. Two theta de seda. Yes. And I'm gonna end up moving out this one half, so I just want to integrate. So this is going to become 3/8. And so now when I go ahead, integrate this. So this is going to be theta minus, and then a coastline integrated is going to be sign, and then I have to integrate the two. So it's gonna be one half from pi over 2 to 0. So if I look at this, I have 3/8 and then I have data, which is pi over two minus two times pi Over two is pie. Sign it. Pie happens to be positive one. So I'm gonna have minus one, okay? And then we want to do minus zero minus sign of 00 So minus one half Mhm. So I'm gonna have 3/8 times pi over two. Yeah, and then I'm sorry. This So sign a two times pi over two is gonna be pi sign of pie is one. So this is gonna be minus one half. So then, if I look at this, I have minus a half and plus a half. So those cancel. So I'm left with 3/8 times Pi over two, which is going to be three pi over 16 Yeah.

To find the boundary for by X belongs to -1. Mhm. You do it X squared equal strips plus two. Some place X squared minus explaining to equal to zero. Yeah the bountiful my between the two given curves and the mars becomes and equals to integration -1- two. Integration X squared takes this to K. X. Squared beaver. Big not solving the riddle. They get integration -1-2. Okay excuse -2 x squared minus key export for big. Which can be simplified and we'll get -177 appointment ticket. Next we find a mix and endless So M. X. equals to integration -1-2. Integration X squared express to the X squared back be very dear. Now solving the inner interval be good. Yeah Integration I guess one good too exploring food. Let's vote. Excuse just 40 square minus explosives upon. To what if we go to 531 17 people. Okay. No girl bloating and I'm very equal to Mhm. Integration -1-2. Integration X. Squared to express to K. X. Cubed. By solving the inner and middle. It will become Okay. Integration -1-2. Explore for just to excuse minus explore face the and it's equal to 18 Upon for you. Yeah. China's defender of the mosque mm minus excessive in coma minus the individual Upon 1 77. And the mosque is and the masses -177 upon 20 K.

Salva before circular phenomena. Diese fourth file Circle of reduce A. We have a part of a circle like these radios. Okay, Have here Yes. X and the y axis on, uh, you know, as a constant. Just constant. Basically. Yeah. Oh, grow so of the month. Good night. Will be cultural force Fourth of the area of a circle. So five i m c squared were four. And density and stroke. That is Emma's elective repute. They want me to finish a book X. Why with these, um really often a graduation about, uh, the access. So So you have got these previous over off Jewish in about the x axis Satisfies the situation on another this one. But this one is that operation also first, let's compute is from numbers. So for that, we can I see that is gonna be equal toe integrate. Okay, inside, off the region D is this vision? The first event, the white squared. Wow, that was the density. The differential by. So, um, so here, um Well, uh, it is easier to me this integral Lena polar coordinates. So here the differential very f d a. Is Are you are the syrup on the world. Why is able to our sign off? Yeah, well, for this vision or angle, Sierra goes between Syria on on. Uh, yeah, my house on Ara goes from zero off. Thank you, heart. Perhaps you're old. Okay. To a through this intro of the well, white squares are signing off, you know, Or a sign up there are squared. That's the fancy. I mean, the the PR Are you set up when the alarm goes from zero A. Terror goes from zero up to five. Perhaps. So we have all the would be integrating first. Er so we're integrating r squared. I'm sorry. So our cube you are into all of that to the fourth power with about four. So that evaluated between agents here would be 84th power. You worry about it, for there is, um, role. Times are things are from Syria occupied halfs off sign square. Yeah, there is this Trigon which paid entity that tells us that sign a square is equal toe one minus, Of course, off to Sarah. Everybody by two. So it would be integrating. Um Oh, God. What? Because your fight pops on the job of these, uh we have there are forced and stroll very well form. So these would be so it's the fourth room. Got about 400 help, you know, the jewel of minus cause I know who. Okay, I don't see what have also hops on the anything control of that would be a sign off you, Sarah. Invited by four today from the moment that evaluated by halves on zero. So this would be equal toe from his fourth flower. Don't throw. I'm so I have the very body that's 5/4 stop by fourths minus will sign off by that issue. No sign of why. A few times by houses signified cases zero the well that minus zero minus sign of zero. That is also zero. So this would be just this part. That is, um bye. Them say fourth forward. I'm through. Invited by, um, or squared. This is, uh, the moment of finished. About six axes now for forms of image about y axis. You integrate? Yes, I don't be X squared. Drew the eight on the X is equal toe are course. So this would be integral from, uh, well are sine theta. It's clerk. And they are are there on our goes from zero to a it affirms your fine house by house. So well, either interest the same here as in here. Drinking are here. You get, uh, what? We have the density. So growth time stage to the fourth power. Do I like four? But then we're left with being to roll from zero up to the firehouse. Off course. Sign. Okay. Don't squared. Oh, yeah. So because I know that the square is about to Yeah, one plus, of course, to Sarah. Very light. 22. And so, um, so these from your off my hands data So they seem to know it's gonna be equal toe throw thumbs. Mm. Divided by for a forced by four. Then develop over one half is their hearts. Plus, uh, of internal What off course? Earn, uh, in jail. Of course. Sign. Sign off.


Similar Solved Questions

5 answers
Identify the region whose area is the limit lim"4[0)+4
Identify the region whose area is the limit lim "4[0)+4...
4 answers
Quettion Cornparr thcVrcumanrahsortion rrollowinoECitonds1650 cin14.0441Dcorerinc DryCC amoweTesonarce stnuctumesGeinannaeelstine Ie QueslLncr vol #ill leam ~hileZatn nointa busedJotanba PouanedFaln Rfeu arlcm +
Quettion Cornparr thc Vrcumanr ahsortion r rollowino ECitonds 1650 cin 14.0441 Dcorer inc DryCC amowe Tesonarce stnuctumes Geinann aeelstine Ie QueslL ncr vol #ill leam ~hile Zatn nointa bused Jotanba Pouaned Faln Rfeu arl cm +...
5 answers
Give RIS designations for all the chiral carbons (if any) in the following molecules Indicate whether each molecule is chiral or achiral:OHOHCH; HO OH CH;CH,CH;CH; CH;
Give RIS designations for all the chiral carbons (if any) in the following molecules Indicate whether each molecule is chiral or achiral: OH OH CH; HO OH CH; CH,CH; CH; CH;...
5 answers
3,6_ Jvlormnoh 1 1 V 1 116 3 HW Scorc: 44.175, # 1 ZE 1
3,6_ Jvlormnoh 1 1 V 1 1 1 6 3 HW Scorc: 44.175, # 1 ZE 1...
5 answers
Question Findimplicit sin differentiation 3 15 { the following equation :f
Question Find implicit sin differentiation 3 15 { the following equation :f...
5 answers
A multiple choice exam is given. A problem has four possible answers, and exactly one answer is correct_ The student is allowed to choose a subset of the four possible answers as his answer. If his chosen subset contains the correct answer; the student receives three points, but he loses one point for each wrong answer in his chosen subset: If the student just guesses a subset uniformly and randomly what is his expected score?
A multiple choice exam is given. A problem has four possible answers, and exactly one answer is correct_ The student is allowed to choose a subset of the four possible answers as his answer. If his chosen subset contains the correct answer; the student receives three points, but he loses one point f...
5 answers
QUESIIOA1[12 marks]EvaluateX+1 xy dydx1[3 marks]Given the region R bounded by y = 3x2 _ 2 and y = 2x.Sketch the R region:marks]Using the vertical slice technique. determine the volume of R region if flx,y) = x+y. marks]
QUESIIOA1 [12 marks] Evaluate X+1 xy dydx 1 [3 marks] Given the region R bounded by y = 3x2 _ 2 and y = 2x. Sketch the R region: marks] Using the vertical slice technique. determine the volume of R region if flx,y) = x+y. marks]...
5 answers
Point) Identify the type of quadric surface defined by the equation2x2 + 2y2 _ 22 = -4,and find all X-, Y-, and z-intercepts of the resulting graph. Sketch the graph of this quadric surface on paper:The quadric surface is a an hyperboloid of one sheet x-intercepts when x y-intercepts when y and z-intercepts when z 1/2withEnter your answers as comma separated lists, or enter NONE if there are no intercepts of a particular type:
point) Identify the type of quadric surface defined by the equation 2x2 + 2y2 _ 22 = -4, and find all X-, Y-, and z-intercepts of the resulting graph. Sketch the graph of this quadric surface on paper: The quadric surface is a an hyperboloid of one sheet x-intercepts when x y-intercepts when y and z...
1 answers
Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\{\begin{array}{l} y=x-1 \\ y=x^{2}-6 x+9 \end{array}\right. $$
Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\{\begin{array}{l} y=x-1 \\ y=x^{2}-6 x+9 \end{array}\right. $$...
1 answers
Disks $A$ and $B$ have a mass of $15 \mathrm{kg}$ and $10 \mathrm{kg}$ respectively. If they are sliding on a smooth horizontal plane with the velocities shown, determine their speeds just after impact. The coefficient of restitution between them is $e=0.8$.
Disks $A$ and $B$ have a mass of $15 \mathrm{kg}$ and $10 \mathrm{kg}$ respectively. If they are sliding on a smooth horizontal plane with the velocities shown, determine their speeds just after impact. The coefficient of restitution between them is $e=0.8$....
5 answers
A child who cries when his mother departs and smiles and runs tohis mother when she returns is displaying which type ofattachment pattern?(A) Avoidant attachment(B) Ambivalent attachment(C) Disorganized attachment(D) Secure attachment
A child who cries when his mother departs and smiles and runs to his mother when she returns is displaying which type of attachment pattern? (A) Avoidant attachment (B) Ambivalent attachment (C) Disorganized attachment (D) Secure attachment...
5 answers
JI Aid3) Evaluate each sum (use the formulas discussed in class). a) 1+3+5+7+9+...+101 b) Find the sum of the first 100 positive multiples of 3. 65 2 (1 220)brarian
JI Aid 3) Evaluate each sum (use the formulas discussed in class). a) 1+3+5+7+9+...+101 b) Find the sum of the first 100 positive multiples of 3. 65 2 (1 220) brarian...
5 answers
In the 3-month period November 1, 2014, through January 31,2015, Hess Corp. (HES) stock decreased from $80 to $64 per share,and Exxon Mobil (XOM) stock decreased from $96 to $80 per share. Ifyou invested a total of $28,640 in these stocks at thebeginning of November and sold them for $23,360 3 monthslater, how many shares of each stock did you buy?Hess Corp. (HES) stock__sharesExxon Mobil (XOM) stock__shares
In the 3-month period November 1, 2014, through January 31, 2015, Hess Corp. (HES) stock decreased from $80 to $64 per share, and Exxon Mobil (XOM) stock decreased from $96 to $80 per share. If you invested a total of $28,640 in these stocks at the beginning of November and sold them for $23,360 3 m...
5 answers
A lens maker is asked to manufacture (a) a diverging lens of focal length 40 cm; and (b) a converging lens of focal lengths of 80 cm. Find the radii of curvature of the surfaces for the two lenses Assume the two surfaces of each lens have identical radii (but not necessarily the same for the two lenses) The refractive index of glass is 1.5_
A lens maker is asked to manufacture (a) a diverging lens of focal length 40 cm; and (b) a converging lens of focal lengths of 80 cm. Find the radii of curvature of the surfaces for the two lenses Assume the two surfaces of each lens have identical radii (but not necessarily the same for the two len...
5 answers
Perform the indicated operation simplify: Express the answer in terms of / (as complex number)9i) (7 | 9i)
Perform the indicated operation simplify: Express the answer in terms of / (as complex number) 9i) (7 | 9i)...

-- 0.019948--