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A) Evaluate II: (c+y +2) dV , where E is the solid in the first octant that lies under the paraboloid 2 = 4 - 2 y? b) A solid E lies within the cylinder z2 +22 =l,t...

Question

A) Evaluate II: (c+y +2) dV , where E is the solid in the first octant that lies under the paraboloid 2 = 4 - 2 y? b) A solid E lies within the cylinder z2 +22 =l,to the left of the plane y = 4 and to the right of the paraboloid y = 1 -r2 22 . The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E c) Evaluate J 22 + y2 dV Where E is the region inside the cylinder 22 + y? = 16 and between the planes 2 2 and ~

a) Evaluate II: (c+y +2) dV , where E is the solid in the first octant that lies under the paraboloid 2 = 4 - 2 y? b) A solid E lies within the cylinder z2 +22 =l,to the left of the plane y = 4 and to the right of the paraboloid y = 1 -r2 22 . The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E c) Evaluate J 22 + y2 dV Where E is the region inside the cylinder 22 + y? = 16 and between the planes 2 2 and ~



Answers

Use cylindrical coordinates.

Evaluate $ \iiint_E (x - y)\ dV $, where $ E $ is the
solid that lies between the cylinders $ x^2 + y^2 = 1 $ and $ x^2 + y^2 = 16 $, above the $ xy $-plane, and below the plane $ z = y + 4 $.

Okay, So this problem wants you to evaluate the triple integral of exposed wide Losey over the region off E. Where is is a solid that is, in the first Occident and is below the pair up. And it is below the probable oId Z equals four minus x squared, plus y the evil four minus x four miles y squared. All right, so the first thing we do is to convert everything into cylindrical coordinates. So if we were to convert our into grand into political coordinates, we would have extra swipe a Z where X is our co sign down our coastline. Dana, we have Why is our side data our scientist? No. And we have the just the and D V in cylindrical cornices are times are times DZ, pr di fada. All right, so now let's look at the boundaries. Yeah, in the first. Often that means our X y and Z components are all positive. Traveling really mattering cornets and our problem oId z before my eyes expire, Biswas, where will that can be rewritten as e equals two or minus r squared? And this is a downward facing problem. So looking at our boundaries versus DZ. So the bottom of our Z is the X Y is the X Y plane because I remember one of our arm. It is in the first octave. So x Y plane is The downward plane is a plane of Z equals zero Essentially, that restricts the region. Then we have the downward facing parable oId Ziegel. Four mice are squared, which is the upward restriction. So we could just let in before my eyes are square. Yes, we have the are So er and this obviously this largest one c equals zero because is done, we're basing problem. So we just said 0 to 4 minus r squared we can get our off to So our radius goes from 0 to 2 and finally our angle theta d'hara. It goes from zero two pi house. Now why is it pie house? Well, it's pie has because is in the first oxygen. So when the X y plane, the X and Y components have to be positive. All right, so now that we've got everything is little coordinates and we've got our boundaries now it's time to calculate it. Okay, so let's first integrate this with respect to z So our boundaries or the R and D d and I say the same. So if we integrated with respect Dizzy Well, I knew it. First get R squared the equal sign data our squares equals and data plus our square and the science data whispers he signed via was 1/2 rz sward 1/2 our a z squared And the boundaries are, of course, from four minus R squared from 0 to 4 minus are square and the other two or d r d Fate up. Okay, Next if we plot If we were to plug in the points for Mayes R squared and zero into our into ground would get the double into grown off double integral of R squared times four minus R squared. Course I do. You know waas R squared. I was four minus r squared. Sorry, they are waas. Well, it's 1/2 are times four minus R squared squared DRD data. Okay, so now we have to integrate. Now we have to integrate this with respect to D I or d thing up. Well, looking at this equation, you can probably tell that this would be This will be a struggle to integrate with respect to our So how about we try integrating it with respect to defeat data first? So let's swap the boundaries to detail. They are because if we did, with respect to D data and, er we're bound to get since we have a co sign and a sign when we're dealing with a region boundary a pirate, too. We're about to get a zero somewhere, so we'll probably end up most nicer than if we just did this ugly Integral will just back to our So if we were to integrate this with respect to data would have an integral from 0 to 2 to the boundaries of our we would get into grown r squared. I was four miners r squared. I'm silent. Data minus R squared times four minus R squared coastline data plus 1/2 one house are times for monies. R squared squared times stayed up and this is from pi over two Priore from zero to Pi Overton Oh, respected data with respect to a d. R. So now if we were to plug in our boundaries, uh, from zero department tomb, we would end up with an integral in the winter grown off of our square Four minutes or square four miners are swear plus pi over four times are times four minus R squared, squared, plus are square All right, four minus R squared, squared because off the coastline so would be at this. Oh, yeah, and respect the d r When were you just being combined? These two terms. And so we were to combine those term we could get a simpler into grow from 2 to 00 to our square I was four minus are square plus pi Ford's Our times four minus R squared, squared Respect the d r. All right. And if we were to expand this out so we can further simple fire we get an integral you go from two from zero to off Oh, yeah and be distributed to r squared into parenthesis east into our before my eyes are square Viet eight R squared When it's to our to the fourth to our to fourth plus pi over four our times 16 are minus eight R Q plus our bit to the fifth power Do you are? And now that we have this, we can integrate with respect to our to get to get in. Over three are cute. Minus to are to the fifth over. Five plus hi over four times IMEs times eight 18 r Squared. When is to argue before? Plus our six over six from 2 to 0. Okay. Wait. I see my hair here. This are not supposed to be there. I already distributed the are. So if now if we're simply plug in the values from zero to, we get a final answer of woman 28 over 15 plus a pi over three. And this is your final answer.

In this problem, we have to evaluate the mask off the laminar that is M equals two double integral Sigma Delta, not DS, where Sigma is the portion off parable oy. That is to that because toe X squared plus y square inside the slender X squared plus y squared is equals to aid. Now we can apply the formula double Integral Sigma fo x y that DS equals toe double. Integral are F o X Y g o x y under Rude, Carl's Ed or colleagues Whole square plus calls it or call viable square plus one deer with that equals toe G o X y equals toe X Square plus y squared over two and f of x y and said equals tau delta, not which is constant. Then we have call off dead or call off X equal to X and call off said over color by equals toe by therefore we got em equals toe double integral Sigma Delta North DS equals toe double delta nort doubling Tingles are under X square plus Vice square plus one D A equals tau delta, or double integral are under root. X squared plus y squared plus one d v. D X on converting toe Polar coordinates using X equals toe are assigned pita and white walls. Tow R sine theta. Therefore, X square plus y squared equals toe are square. The Tita varies from zero to Dubai and 02 under good. Varies are before the above NT girl converts into Delta Nord double integral from 0 to 2 pi and zero toe under root age under root R squared plus one r de are de Tita, not substitute r squared plus one equal toe Be on to our d are equal toe dp We got Delta nod over to double integral 0 to 2 pi and one toe nine under o T d t d. We got Delta in art over to integral 0 to 2 pi and evaluating the limit to buy three My little i p to the power three by two Tito Cita de Tita We got Delta and Art 27/3 minus 1/3 integral 0 to 2 pi de theater Now evaluating the limit, we got dealt an art nine minus 1/3 multiplied two pi and finally we got 52 by Delta Nord over three hands the mass off the cob. Lemina is equals. Tow em equals toe double integral Sigma Delta Nord B s equals toe 52 5 Delta north. Over three. So that's the solution.

So in order for us to find the volume between these two, what I'm gonna do to start is just draw a sketch of what this shape might look like. Now, a private Lloyd is going to look just like a quad track and then you, like, rotate it around in three dimensions in some sense. So if we were to plug in 00 for X and Y notice, we get out 24. So it's going to be like up here, and then since both X and why you're negative, that means it's always going to be decreasing from this point. So look, something kind of like that and then it just kind of wraps around. Okay, Now for our cone over here, Since this is the positive root of it, we know that all of the output should be positive. And if we plug in 00 we get the height of zero. So we're going to be starting here, and then it's just going to, um, come up like this and then it would just keep on extending forever. Well, we also have this little intersection here. So now what we're interested in is all the volume in between these here. So I mean, probably not a good drawing. But all we really care about from this is that the red function is on top. The green is on bottom or the problem is on top and the cone is on bottom. Now, since they want us to do this in cylindrical coordinates, we want to set up our volume integral to look like this. It will be some bounds for data. Ah, some balance for our some balance for Z. And then we'll have our DZ d r d theta mhm. So we want to take both of these first two polar coordinates so we can solve for what is r and data. And then in that same process, will be able to figure out what our balance for ZR Um, just by, um, what's the word? Changing these two polar coordinates as well. Now, the substitution that we're going to want to make I'll make things open easier is R squared. Plus y squared is equal to r squared or X squared plus y squared. And I guess that export plus r squared. So if we do that over here for a problem, I'd will get Z is equal to, um 24 minus r squared. And then over here, we get to times square root of our square, which is just going to be our And in our case, we're going to restrict our to just be positive. So we don't need plus or or the absolute value of our there. Yeah, so this is going to be our lower bound to our and then up here is going to be an upper bound 20 for minus R squared. Yeah. Now for are mhm. Uh, excuse me. So, to find our we can go ahead and set these two equal to each other. Um, so we can find what the upper ballot for is going to be. Because notice that, um, first for theta this is just going all the way around is complete rotation. So that would just be zero the two pi. So we actually don't have to work too hard there. Um, but now to get the data, I mean, are we going to just set these two equal to each other? And so doing that, I'm going to move everything over to the right side. It's going to be R squared plus two ar minus 24 is equal to zero. And then we can factor this to be our minus or ar minus six is equal to zero. And then this is going to be our is gonna four or uh, are plus six here. Are you going for R is between negative six, but in this case, R radius, we're gonna just restrict it to being non negative. So are lower bound is zero because we're starting from the origin and then we're just going out in this direction here. So then that would mean we throw out negative six and our upper bound is going to be four. So now we know what to integrate. And then from here, we can just go ahead and integrated. So one thing to notice is that this integral depending on theta well, we don't have a theta anywhere, and the first to enter girls also don't depend on data, so we can just go ahead and pull that out. So it be 0 to 2 pi of one essentially D data. And now we can just focus on this inner integral. So are is a concept of prospective Z so that would give us 0 to 4. Um, rz evaluated from two are 2 24 minus r squared D R. Now, if I integrate one that would get data evaluate from zero to buy, it would just be to buy. Now, here. If I just multiply both of those by our that would be so. First interval from 0 to 4 of 24 ar minus are cute. Minus two are squared t r And then let me just make sure I did that. Right, Um, 24 or R cubed? Yeah, that looks good. Now we can go ahead and evaluate this integral. So would be to pie times. So we just use power rule. So B 12 R squared minus 1/4 are to the fourth and then minus two thirds R cubed, evaluated from 0 to 4. Um, So if we go ahead and plug in four into this, um, we should get 256 over three, and then plugging in zero would just get zero, so multiply that by two, and that should give us 5 12 pi over three. So this is our volume or part? A. Now, um, for part B to find the central with a constant density. We're just going to use this equation here where T bar is supposed to just be the center for whatever variable we're using. I just used me as a stand in variable. Now, one thing if we come back up here and look at this no, I didn't draw this very well, But if we think about it just in the XY plane, I was talking about how this is just a circle. So since it is a circle in the XY plane and it's pretty symmetric if we were to kind of look at it, we have as much in each of the four quadrants for the X Y plane. That means the central for axe And why are both going to be zero? So just to save us a little bit of time and calculations, we're just going to say, since our shape is symmetric in terms of X y plane, then X bar is equal to zero and why bar is a good or zero due to constant density. And so this constant density is important because if it wasn't, uh, then we'd actually have to work a little bit harder, unfortunately, Um, yeah. So let's go ahead now and just plug everything in. So we already have the volume, so that would be 5 12, by third. So let's just do that to be the bar is equal to. So it would be the integral of so it would be 0 to 2 pi integral from 0 to 4. Uh, what was our bounds for our again to our 2. 24 minus r squared who are 20 for minus r squared. And now we have to have tea, which in this case, is going to be easy. So it be easy. And then our volume element, remember in cylindrical coordinates is supposed to be d or are DZ d r d theta. Okay, so now we just need to evaluate this, and then we'll be good to go now. Like I was saying, before this d theta here, nothing in between depends on data at all. So we can just go ahead and pull that out, evaluate that by itself and again that will just give us two pi and then that gives us more time to just focus on this. So first we have integral from 04 Now, integrating this with respect to Z is going to give us one half z one half c squared R evaluated from two are 2 20 for minus R squared, and then we have d are on the outside. So if we were to actually, before we do that, notice this one half and this to cancel with each other now off on the side, I'm actually going to go ahead and figure out what, um, z Square is going to be. So to be 24 minus r squared squared, and I'll just put our and then minus Ah, so that would be or r squared. So for r cubed. And now we can go ahead and expand that. So we are, um 5. 76 minus 48 r squared plus are to the fourth and then minus four are cute. Um, now we can just go ahead and at our like terms after we distribute the are so to give us 76 r and then this becomes r cubed. Combined it with this one. So b minus 52 are cute and then plus are to the fifth. Yes, that's what's going to be on the inside over here. So let's write that out. It would be not to pie. Just pie. So pi integral from 0 to 4 of 5. 76 ar minus 52 are cute plus our to the fifth Ah d r. And now we can go ahead and use powerful to enter each of those would be pie now 5 76 divided by two, some six divided by two us to 88 r squared. And we're just using powerful to integrate. These, uh, 52 divided by four is 13. So 13 are to the fourth, and then +16 are to the six. And we evaluate this from 0 to 4. So if we go ahead and just plug four in, uh, we should get, um, 5888 over three. Hopefully, I didn't mess up plugging anything into my calculator. And then, um, just zero. So that means, right. Is this Actually, Actually, I made a slight mistake. This isn't Z Bar, because I forgot to divide this by the volume element. Let me just a race that and pretend like I didn't write nonsense. Um, yeah, So now we need to come up here and get our volume element and then divide that and then that will give us our center of mass with respect to the easy access. So now we're going to get Z bar is equal to. So this 5888 over three pi divided by by 12/3 pots. So first pies canceled the threes, cancel. And then, um, let's see if that is divisible by 5. 12. So that actually gives us 11 0.5. Yeah. So that means our central roid, as long as we have a constant density, is going to be 00 11.5. So that this is part be up to them. I just go ahead and scoop this down. Also, just kind of apartheid there as well.

Okay. So we have a it's a cruel world over E. Of X minus Y. T. V. And he is the region between the two cylinders, X squared plus Y squared to go 16 equal to one expert post by square 16. I love the X. Y. Plane. And below you love him. Z is equal to fly. So we know the X. five planets disease equals zero. So we have to bounds on Z. And then balance on X. And Y. We know that it's cylindrical. These are just are squares and that turning Z. And she was this cylindrical which gives our science data what's for. Okay so we can go ahead and start converting his entire integral into cylindrical. So we know that um Mhm R squared is between 16 and one. So that just means ours between four and 16 and four and one. We know that sees greater than zero and then Z is less than or equal to our science data. That's four. Yeah. Okay. So then we just need to find potatoes where this is um yeah valid. So notice that if we have in our where are goes up to four. So this max value is four. We know that science data's max value is negative one. So we can see that. This can never be zero, this can never be less than zero. So we don't have to worry about anything as we can just go ahead and Do this from 0 to Pi um 124 and then 0 to our science data plus four. And then we know X minus. Why would just give us our we have another and another R. Since T. V. Is equal to R. D. C. T. R. Data. Yeah we have R squared times. Co sign paid up on assigned data. Tze already data. Mhm. Right. So now we can't separate these intervals yet. Just because we have datas and datas and ours all mixed together so we can go ahead and just start from the middle. We have no Z's so we can just integrate see so we have our squared co science data. Science data times are science data. What's four? And then might receive that. Which we don't have to worry about. We still have interlocked data's in dollars so we can separate it yet as well. And if we go ahead and just do a quick distribution we get R squared um Times are science data codes and data Plus four Co Cynthia. Yeah minus our side squared. They know. Mhm -4 Side Data. Hey are you data? And then again to get our cube sign data, coastline data was for R squared coincide data minus our cube science where data -4. Our Spirit Side Data. I PRT theater. All right. So we can start integrating inside giving us arts to 4/4 signs. They do because I data plus this is for our squares we have four thirds R cubed whose side they're uh Minour are to the 4th over four sine squared of data And then 4/3 are cute inside data. All right. Then we have to evaluate this From 1- four and then detail. Thank you. Right, okay. Mhm. So there are a couple into goals that we can go ahead and start crossing out. We know that we're eventually going to get constance as our coefficients and then we'll just be left integrating co signs or signs. We know that the integral over syria soup. I focus on data data and this is the same as he had to go from zero to pi of science data potato. Both are equal to zero, which means that this term in this term Just go to zero and we can go ahead and yeah just write this again As the hearts of 4/4 science data consult era minus Arts of 4/4 sine squared data. From 1 to 4 data. Yeah And then start evaluating this. So we have your integral from 0 to 2 pi We have four we have 16 which is um sorry we have four to the 4/4 minus 1/4 1st. That will be the coefficient of both terms. Which is just For Cube -1 4th or 64 minus 24 Gs 2 55/4. So we have 255 over four which is the same for both times side data, coastline data minus sine squared they don't oh data yeah we can go ahead and Pull out the 255 over four and integrate. We know that um She from our double angle identity. That sign of two data is just two sides data. Cosign theta. So this quantity here is just um sine two theta two minus sine squared data. Yeah. Mhm. Mhm. Your data. We also know that sign square data can be reduced using the double angle identity and then we can go ahead and do that as well. Okay so that if we go ahead and just flip the signs we received minus one half plus co assigned to data over to. Yeah. Yes the data and before we see we have a sign and a co sign from an integral from zero to pi. So we can just crossed his out since the angle was doubled. It's still a periodic integral. So then we just finally end up with 2 55/4 times the integral, from zero to pi negative point half to which just ends up being negative to 55/4 times pi as our final answer.


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