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Problem Let R be the quarter in the first quadrant of the disk with radius R centered at the origin: Find the center of gravity (7.J) of R: Use polar coordinates an...

Question

Problem Let R be the quarter in the first quadrant of the disk with radius R centered at the origin: Find the center of gravity (7.J) of R: Use polar coordinates and cange of variables in the integral. Problem 2 Consider rectangular craboard box without top aucl hottom. The dliagonal of the box has leugth Use Lagrange multiplicrs to findl the maximum CurhceH of the [mpeT uUsedd t0 make this hxx. What ATe the ditHsicx of this optital hx"

Problem Let R be the quarter in the first quadrant of the disk with radius R centered at the origin: Find the center of gravity (7.J) of R: Use polar coordinates and cange of variables in the integral. Problem 2 Consider rectangular craboard box without top aucl hottom. The dliagonal of the box has leugth Use Lagrange multiplicrs to findl the maximum CurhceH of the [mpeT uUsedd t0 make this hxx. What ATe the ditHsicx of this optital hx"



Answers

Evaluate the given integral by changing to polar coordinates.

$ \iint_D x^2 y\ dA $, where $ D $ is the top half of the disk with center the origin and radius 5

Were given integral of a function over a region and the description of this region were asked to use polar coordinates defying this integral, the integral is double integral over the region de of the function X squared Y d A. Where d is the top half of a disc with a center at the origin and a radius of five. So we can draw our region de using polar coordinates as follows. We have a line that equals pi over two in line fate equals zero, and we knew that maximum value of our is going to be five. We have that. This is the top half of the disc, so we're only looking at data from zero supply antes. It's a disk we're looking at. All radius is less than or equal to five. Great, there equals zero. So our region de looks something like this this blue shaded region which is a semi circle, and we see that and pork ordinance the description of the is set of all ours and potatoes such that our last between zero and five and data lies between zero pie. If we treat our function X squared y as some function of X and Y call it f of X y. We see that X square is a continuous function and why is a continuous function on de and therefore F is continuous on the region de And so since region he is simple It follows that they don't want to grow of x squared y over the region de is well defined and is given by the integral from zero to pi integral from 0 to 5. Uh, polar coordinates X is are who signed data. This is our assigned data squared And why is r sine theta This is or sign data and then for polar coordinates D A is when you are, er data Where the are is that it? In the change of variables this is equal to And since we have that the limits on the integral our constant values We can separate these inter girls using Roubini serum or even his product ID in jiggles. So this is the same as the integral from zero to pi uh, who signed square data sign Peter de Theater times the integral from 0 to 5, uh, are to the fourth er and using substitution for the first integral say u equals co sign of data. Let me have that. Do you is going to be negative. Sign of data de una and therefore and a girl is equal to integral from you have zero is co sign a zero which is one to you of pie. Just co sign a pie which is negative. One of negative you squared Do you times integral from 0 to 5 of art Fourth, you are is equal to two steps at once. So applying the negative we conflict the limits on the integral and then also taking anti derivatives We get 1/3 you cute evaluated from negative oneto one and times one fit part of the fifth diverted from 0 to 5 is equal to 1/3 time's 1/5 is 1/15 times wanted to third is one minus negative one that there is negative one. So this is to times 5/5 is five to the fifth, minus 00 So being five, this is equal to two times five to the fourth over three. Five to the fourth is the same as 25 squared. So 25 times 25 This is 625. So we get 625 times two over three, which is 12 50 divided by three. This is our answer.

Okay, so we have, um the double integral over the region are of e to the negative square root of X squared plus y squared d A and our region are is to be described by the disc from the originate from the inequality of X squared plus y squared is less or equal one. And let's just draw out this region are really quick xto Why, If we look at the boundary of this region exportable. Spicer is a little one. This is basically just the unit circle at the centered at the origin you this circle. Okay, just do it this way. Perfect. And then says his listener, equal to its anywhere inside this circle so this would be our region. Ah, Now we just need to write this doubling blase polar The winner. Goal e then. So, basically, um, you just, uh, to change this rectangular component right here to Porter. You just remember that X squared plus y squared r squared? We substitute that in, and then we multiply this by the Jacoby in art yard feta. And then we need to set our limits for our and data for our We just think about radial lines or change from the origin and going outwards. And we see that the radio strangers from 01 and then it does this over 0 to 2 pi. Okay, so that would be our double integral. Pretty simple. Um, And now let's just solve for this stuff on your girl. So if you notice the anti identical is entirely are integral and then the outside integral is just a data interval. So we respect to the data integral this inside integral. It's actually just a constant numeric value. Soak after you slide the side to the front 0 to 1 of each and make of our times, R d r and then 0 to 2 pi of data. Basically, what I did was I transformed a deline Urdl into the multiplication of two single intervals, which is pretty neat, in my opinion, and it also makes it easier to solve. So this part is just where it makes it less complicated to solve because you can separate it out into its components and self each part more cleanly. Um, the second part, or the second goal is just to pie because single integral of function one is just the length of the interval And then this first part we're going to need Teoh Use integration by parts, so let you equal are and let TV equal each of our t r. And if you're confused, why I choose you to be equal to our just remember the acronym lie eight basically lager them in verse. Trig algebraic trick on an exponential. That's the order in which you want to choose your function You to be okay. So since we have you evil toe are do you would be equal to d R And then be we just take the anti derivative respect toe are the right hand side, which would just be each and egg barred by by one. Um, I'm just using the trick that this is a linear function inside of exponential, and we just take the anti driver of the exponential like usual. And then we just divide by the linear coefficient and this works for any other complex function that a linear function may be embedded inside, So it just ah, little shortcut. That is pretty handy. Um, okay, then basically say that this entire thing is equal to U times v. So part times negative eats a negative are from 01 minus rule from 01 of e Do you so negative Each and negative are and then do u S d r Um, Okay. And then we just evaluate this, that we get negative one use a negative one, and then my eyes what we get when you plug in Syria, which is just zero. So the first part, it's just that and plus each of negative are over an IQ of one just using that little linear trick as we used before. Um, this should be equal to night one over e plus we get when we plugged in one, which is just, if one over e and then what? We going plug in zero. Which would just be so we do minus. And then, since this would evaluate Tunic Wonder just a plus one, this is equal to negative two over e and then plus one. And then we just need to multiply by two pi this entire thing. So, yeah, on that should be our final answer.

All right, So we're giving a political or region are other function e to the negative X square minus y squared d a, um such that the region are said to be described by the disc with the inequality of squared plus y squared is less than or equal to four on. Let's just draw up this region are really quick if we look at the boundary really quick, so extra tickets y squared is equal. Before that would just be the circles turned out the origin were afraid. Yes, to so strong this circle there 1 to 1 want. And then basically, since this inequality is less than or equal to its basically anywhere inside fetus or along to just said this would be our region are as we need to write this in polar coordinates or this interval important for corn it. So let's do that. Um, if we look at, uh, this exponent year, um, it's not a traditional X plus y squared, but we can factor out negative to make it look like a surplus of ice cream. And then we know what X squared plus y spirit does Polar cornets just r squared. So we're gonna go to that r squared negative r squared and then arj Akopian is RDR data. And now we want to think about how our data range for our think about radial lines protruding from the centre and going outwards, and we noticed that it ranges from zero to to basically the outer edge of the serval. And then it does this for 0 to 2 pi, which is our angle range or theater range. Okay. And this would be our integral that we need to sell for. Um pretty simple. Ah, what I'm gonna do first, though, is gonna notice that, uh, the inside intervals that completely are type integral and the outer and roses data a room. So with respect to the fated angle, this entire inside Enbrel is just a constant value. Because in the end, you will, um, evaluate to a new married value so we could just slide this out to the front, says hurted. Two of each and r squared times are the Are the time deserved to pie of potato? Okay, on basically, what I just did is I turned a double integral into the multiplication of two single intervals. It's pretty neat in my opinion, um, and makes it easier to solve, in my opinion to So this second, Inderal is basically just the single girl of the function one and that basically just translates always to the length of the careful. So it's just two pi, which is pretty easy. And then, for the second or the first in your girl, we actually need to use on some use substitution. We're just gonna let u equal Mega r Square are basically the supposedly function, and then or the anti derivative and its derivatives. So this will be the anti derivative, and then our would be its derivative. So we always set you to be the anti derivative, because once we drive it, get negative two r d r Then we can say RTR is equal to negative d you over to and then we can just substituted. So RDR is negative. What happened is gonna pull that 1/2 to the front. Uh, do you? Then you just want to do e to the U and then change our our limits. Say you elements. So once we plug in and serve here, we get zero. And then when we played in two, we get negative four. Um, I don't really like this. So what I do is to split them, get 1/2 in the front and said Negad 1/2 because from negative 4 to 0, usually you do you and then times two pi. So this 1/2 should just be a pie. This should be equal to each of the U. I got 40 sequel to high times we get when we played him zero, which just each Israel, which is just one and minus We get when you play, connect four so you don't get fourth, and that would be our final answer.

It's probably way Wanna integrate X squared y over this region here? D which is a half disc with Radius five center 00 Since its circular, we're gonna go ahead and switch to polar coordinates because it will be easier. So for polar coordinates, we have first have to figure out what our is doing. So are starts at 00 and goes out. So it's starting here at R equals zero and going out here, it's a circle of radius five. So r equals five. And then what stated Doing well, they just starting here at the positive X axis we're going around and ending up here at the negative X axis. So starting here at that equals zero thio Here, data equals pi. All right, then we have to also know that the conversion X equals r cosine data. Why equals R signed data and d A equals r D r d Fada. And remember, Dia is the area of one of these little sectors right here. Okay, so now let's just put it all in there. Double integral x squared, So r squared coastline squared data times why r sine theta times d a r d R D theta. Okay, are on the inside going from 05 Data on the outside going from zero va by. Alright, so let's gather everything up. Here we have our squared. Are are so four ours, or are to the fourth Goldstein Square Data Signed data D theta dinner. Girl of Art to the fourth. I'll be on quips arts. The are to the fifth over five. So now zero pie are to the 5/5 from 0 to 5. Because I'm squared. Data sign data deep data. So we got five to the five of our five. That makes five to the fourth, which is 5 25 1 25 6 25 minus zero and then zero to pi coastline square data find data. Do you data? Since we only have one sign data we're gonna want it to be d you and then you will be the co sign you will be the cosine of data and then d you will be negative. Sign of theta d theta. So to make this work, we need a negative in here. Negative out here. And if they do equals zero u equals the coastline of zero which is one. And then if you equals or if they t equals pi. The new equals the coastline of pie, which is negative one. Okay, so now we're integral. Becomes minus 6. 25. Want to negative one? You squared. Do you? That's minus 6. 25. You cubed over three. Negative one No one. So that's negative. 6 25/3 negative. One cube Negative. One minus one cube, which is one. So since negative 6. 25/3 times. Negative too. So 1250 over three.


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