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Question 710 ptsUse the transformation u = {2 %y, v = Zy toevaluate the integral below on the parallelogram R with vertices (0,0) , (2,0) , (4,2) , (2,2) . Ifr(2y +...

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Question 710 ptsUse the transformation u = {2 %y, v = Zy toevaluate the integral below on the parallelogram R with vertices (0,0) , (2,0) , (4,2) , (2,2) . Ifr(2y + 4zJaA

Question 7 10 pts Use the transformation u = {2 %y, v = Zy toevaluate the integral below on the parallelogram R with vertices (0,0) , (2,0) , (4,2) , (2,2) . Ifr(2y + 4zJaA



Answers

$15-20$ Use the given transformation to evaluate the integral.
$\iint_{R}(4 x+8 y) d A,$ where $R$ is the parallelogram with
vertices $(-1,3),(1,-3),(3,-1),$ and $(1,5)$
$x=\frac{1}{4}(u+v), y=\frac{1}{4}(v-3 u)$

We want today. They're going to solve a problem. About 16 here we can drive. It's a the picture of a battle like you know, so let and one represents what it was. Eight minus three x These are the equation off the line, so a 2% were equals. X Successful Air three represents y equals minus three x l four Represents vehicles X minus four l want Connects one come off five and three column minus one l two connects kind of one. Come on, three and one Come out. Five l three connects minus one comma three and one Come on. My ministry l four connects one common my industry and three comma minus one for l one will be one by four indoor leave minus three You, which is a cause minus Trian do one by four in the u plus v because eight held two equals one by four in do the minus three U, which is equals one very foreign do U plus the plus four Health tree was one by four into V minus. Do you, which is minus trained you one day four and do you plus V and for ISS won by four and Duke we minus three you, which is one by four into you plus me minus four. Using the parametric equations, we get X equals one by four into U plus v. Why equals one by foreign do B minus three you so everyone will be vehicles eight and you will be u equals minus four. A three will be vehicle zero and four will be U equals four. So solving integration for you and we provide the known for the independence Not for less than you. Less than records for zero less than or equal city less than recall toe eight Soto ex way by Do you really? You could one by 16 plus three by 16 one day four Taking partial derivative off X equals one before in the u plus v y equals one by four in the V minus three. You with respect to you and we we get for X plus eight million Toby equals foreign do one by four and do you plus V because 18 do one by four in the re minus three u Do you devi the that you pleasantly plus to re minus 16, which is three v minus five. You in do do you be so in bagels? Zero do it. Minus 4 to 4. Three V minus five. You in the one by four the U. D. V, which is one day for indu 0 to 8 queen three v. U minus. If I buy tow, you square from minus four before dealing which is one by four in the 0 to 8. 24 v David which is one by four in the well, the square zero to it, which is three Indo 64 which is 19 So which comes to the That's and awful question. Thank you.

Today we are going to solve a problem. Numbers here, the Caribbean Matics we need to find given us X equals Do you Plus me directs by the U equals two Directed by do the equals one vehicles u plus two v Don't buy by the U equals one. No, by by dozy equals. So the golden metrics is given by to move on one. Well, so which is equals four minus one, which is three. So this is that patients got a person's graph So now we substitute x equals Do you purposely executes do you plus me and y equals u plus moving in the equations off lengths we get? Why equals two weeks u Plus two recalls to Indu Do you plus me u equals zero then X equals Do like do you plus vehicles two into U plus doing we get vehicles zero then X plus y equals three Do you plus the plus you plus Tau V equals three vehicles one minus you. Then in the European, we can define the region as follows r equals you Come out early zero less than a record to be less than record toe one minus you zero less than or equal to you. Well done. Recalls one. So point, fight one point. Fight one. So this is this 0.0. Come over. One, this is one comma zero. So this area will be vehicles one minus you. Thank you.

Yeah. In this question, were asked to evaluate a double integral that's over a triangle. And were given a transformation. That is the set of variables that we're expecting that we should try and make use of. So how do we go about this? Well, the first thing we should do is we should first find the jacobean of the transformation. This is very important for us to find. So why don't we? So that's basically a set of that's a set of partial derivatives from which we take the determinant out of it. So let's find the partial derivatives first with respect to both U and V. Of both X and Y. And we'll note them down here in green too. We know that the partial derivative of X with respect to you is just to and with respect to V is just one. And the partial derivative of why with respect to you is just one. And the partial derivative of why with respect to fee is too so substituting all of this into our determinant formula so that we can get our jacoby in our jacoby in is going to be the determinant of 21 12 which is just going to be four minus one is three. So that's going to be what we multiply everything by now. Yeah. How do we deal with finding the equations? How do we deal with finding out the region of integration? The region of integration looks like the following. We have a point at the origin and we have a triangle. We know one point is the 1.0.2 comma one and the other point is one comma two. So the region looks something like this. Yeah, so this is our region event, that's our region of integration. And we need to find equations for each of these three lines. Fortunately it's not too difficult to do so because what we have here is this equation here? If we know that that's just why equals up to left one, that's just two X. This equation is up one left to Yeah, which is just going to be, we can say that this is just X equals Y equals X, divided by two, or equivalently X equals two. Y. And we know that the third equation here is just going to end up at three com at zero comma three. If we were to extend the line up one so we can call this equation X plus y equals three. So remember remember that we said remember our transformation we said that you would use X equals two U plus V and Y equals V U plus tv. So we basically do, what we basically do is we substitute all of these into the equations so that we can find our bounds that we want to find. So yeah, here's what we'll do in the first equation we're going to have yeah, we're going to have Y. Which is you plus two V. That will be equal to two times X times two U plus V. Which is going to result in for you Plus two V equals U plus two V. Which implies that you will be zero. Similarly, if I plug them into the second, the second equation X equals two. Why? We're going to have to U plus V equals two times two. U plus V equals two times U plus two V. And what this results and we have to U plus V equals two. You plus four V. Which implies that V will also be zero. Finally, if I plug this into the third equation, that is two. U plus V plus you plus two V equals three. Yeah. Yeah. What do we get in the end? We'll get that three. You we'll get that three U plus three. V plus three. V equals three. And we'll just divide everything by by three and then move the U. To the other side so we can get that V is equal to one minus you. Yeah. So what is our what is our actual region of integration going to be? Well how can V be equal to zero? That will be when U equals one. So we know that our bands of integration if U equals one then V has to be zero. So we know that you will be in between zero and one whereas V will be in between zero and one minus you. Mhm. We can say this so therefore are integral becomes the integral over the region R of x minus three. Y integrated with our differential area becomes an integral from 0 to 1. And an integral from 0 to 1 minus E. U. Yeah. And we're going to have to you we're going to have to U plus V two U plus v minus three times U plus two V. You plus tv all of this multiplied by three D. V D. U. Because you can't forget you can't forget your jacoby in. Okay so what this is is we can bring out the three to the outside. And by expanding this we're going to have minus three U minus six V. Inside the inside the brackets. So by simplifying this we're going to get this, this is equal to three times the integral from 0 to 1. The integral from 0 to 1 minus you will just have minus you minus u minus five V DVD you first firstly we're going to integrate with respect to V because we cannot separate this neatly. This is going to be three times the integral from 0 to 1. And doing this integral in green. We're going to have that. This is going to be eager to minus U v minus 5/2. V squared. Integrated from V equals zero to v equals one minus you. Yeah. And don't forget the d'you hear? Okay, so when I put it in zero everything will cancel. But when I put in one minus you were going to get that? This is equal to minus you times one minus you. Yeah minus 5/2 times one minus U squared. And simply simply expanding this. We're going to have that. This is going to be equal to right U squared minus u minus 5/2 times times one minus two. You plus U squared. And by expanding this again by expanding this we're going to get minus 5/2 minus five. You minus 5/2 U squared yeah mm And don't forget Plus here. So this is going to become in total This will just be yeah yeah minus this will just be in total will have for you for u minus 5/2 minus 3.5 minus three over to U squared. So that's our second integral which means that the integral will come three times the integral from 0 to 1 of for you minus 5/2 minus three over to U squared. Do you? And evaluating this is very easy because it's just going to result in three times two U squared minus five minus 5/4 times minus 5/2. Sorry times you minus one half. You cute going from 0 to 1. Obviously if I put in zero, everything will cancel and what we're going to get after you plug in one. This is going to be equal to negative three. That is the answer to this question.

The problem is used, given transformation. She already begin to grow. You are is a parole. Aground with braces two one three one nine Q three three Nicked one on one off the transformation. This that's going to work, Force House likes you. Asked we Why is able to our force Tom's remind us three years? The first the region are It's funny to buy. Why you going to axe us war? Why he could to Pax plus two why he could to connective reacts. Why could connective reacts half eight? By the transformation we have the region as its equal to you're mean you is between active or connect you one connected to these between zero and it on then partially exp lai. Why should you be if they want to my force So we have to grow r Well, ax Plus it. Why? Hey, it's the control integral from zero to eight into profound negative. Or make two one negative too. And full ax. Ask it why you seek or two three minus for you, Holmes, about a horse you you feel which is controlled by force into girl found zero to eight six re plus already. I mean and the answer is one o it


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