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Use the Change of Variables Formula to compute Js(x + 3y) dxdy, where D is the region bounded by x + Zy = 4, x + 2y = 4,y = 8, y = 12 (Use symbolic notation and fra...

Question

Use the Change of Variables Formula to compute Js(x + 3y) dxdy, where D is the region bounded by x + Zy = 4, x + 2y = 4,y = 8, y = 12 (Use symbolic notation and fractions where needed )f Jskx + 3y) dxdyhelp (fractions)

Use the Change of Variables Formula to compute Js(x + 3y) dxdy, where D is the region bounded by x + Zy = 4, x + 2y = 4,y = 8, y = 12 (Use symbolic notation and fractions where needed ) f Jskx + 3y) dxdy help (fractions)



Answers

$$
\begin{array}{c}{\text { Use the transformation in Exercise } 3 \text { to evaluate the integral }} \\ {\iint_{R}\left(3 x^{2}+14 x y+8 y^{2}\right) d x d y}\end{array}
$$
for the region $R$ in the first quadrant bounded by the lines
$y=-(3 / 2) x+1, y=-(3 / 2) x+3, y=-(1 / 4) x,$ and $y=$
$-(1 / 4) x+1 .$

Were given a transformation and an integral and were asked to use the transformation to evaluate this integral for a region are in the first quadrant founded by the lines. Why equals negative two X plus four y equals negative two X plus seven y equals X minus two. And why equals X Plus one and the integral is the double integral over our of two x squared minus X Y minus y squared. So we have the double integral over. Our first of all notice that are integral can also be written as double integral over our of. And then we can factor out in X minus. Why then we're left over with a two x plus. Why dx dy y and now making the transformation from example one U equals X minus Y and V equals two x plus y. This is equal to the double integral over the transformed region g of u times v times the absolute value of the Jacobean, the X y, the U V do you TV And of course we have from example one. The Jacoby in was one third, so this is equal to one third times the double integral over g of U V D U T V. Now to calculate the sender, Grover want to represent or find the transformed region G. So we have our X Y equations for the boundary of our region are. And from these we want to find the corresponding you the equations for the boundary of G this had before our region are is bounded by the lines y equals negative two x plus four y equals negative two x plus seven Why equals X minus two and why equals X Plus one and using the transformation, u equals X minus y V equals two x plus y. Each of these equations becomes three UV equations. V equals four V equals seven U equals two and U equals negative one. And therefore our double integral is now equal to one third time's. The integral from U equals negative one positive to integral from the equals 4 to 7 of U V. DVD you and using food Beanies. The're, um here. This is a product of integral. So we get one third. I'm just going to skip a few steps here times one half times you squared evaluated from negative 1 to 2 times one half times V squared, evaluated from 4 to 7 mhm and so plugging these values in we get 33 4th This is our and

We will take it to every way to turbo Integral. Oh, this is double integral and distant. Min x x x Cube times Why do you Why your friends? We, uh this our testament are is bonded by just two impressions while you go to extra spare and do I you go toe to eggs you're to everywhere to disturb your integral would like to find the bones. Those founders roar, burn for eggs Throw a bone for why oppa bony for X and proponent For what? So to find this barons beds just a sketch or those to function. You see, this is where I could to excess square. This is a quadratic function where we heard for why I could why go to x x squared? You see that if we heard value X equal to minus, let's say minus two. Why would it be good too home when you were a nerve crazy Implement us one. Why's goto What when we have zero x go to zero wise going toe there When he heard why goto one exclude no one. Why's Goto Why when we have a Y kowtow to excluded for and sword when you have three Here. We have a name. I want a boat. Why? Good toe minus What? No, For a seat. Other boats. This one wouldn't. When we have wash, you could too. Toe eggs seem very key. Organic? Half if we have minus two This We have minus four. If we have remind us one. Yeah, minus one. Why would it be cool to minus two if we have 07 times 00 Why would we go to zero when you have for one? Never have a tool. When you have tool, you have it for when you are three. You ever have six in the so called soul. So? So let's try to sketch those two functions. Maybe her. All right. This is okay. Okay. This is very from pecs. This is why here. Sorry. Here. May story was direct story to where? There. Now it doesn't work. Okay. Yeah. Make it. Does pro fish from my as good as I come when I can extend it here. This book. So Okay, let's catch this function. Size touched. But here? Yeah, we have one. Have to. You have three. You have one. Have tool. You have to Yeah, we have one. I have to. You have three, you know, for Try to think like this. This is a way. This is it. Picks. So you have one? Yeah, we have to hear. Huh? Three have minus one. They have minus tool. You have my last three. Yeah, we have one. I have to. They have three her for good. So just get this first function that make fuss function with brute. When you have negative tour, we have full. You know that negative toe before is some point here You, When we heard of negative four on here is the one means that's the point here for her to zero. We have a point here. If we heard of one and we have the one is We have a point. He we have to We have full Bihar for profit again. Here to form I po int he So you have to draw it away. So when it was try to sketch this function, This is a quadratic function course. Remember that You have to use this how to measure the unit. So is like this look so fresh. I think that's right. Did where? Good? Yeah, or function repressed to zero. This is zero. And then you go up good. The 2nd 1 Let's give It's the girl for bread cake. When you have minus four minus two, we have minus for somewhere. Hey, if we have minus one is minus two somewhere he free her of zero zero. We have one is two one is too. Okay, when we have two were full here when you have to. We have six here, so we'll have a Ryan, which is like, Okay, straight line. So means that what we need the boners we needs is this one. This is too fu. Okay, now we have to find the intersection because the section is this warm friend. How the bones. So in this case, for this terminus, where X is from zero to two and weigh a why is from 0 to 4. So this is the R. This is Ranger for our so means that here, we're gonna have into sugar. He got through into rule French rates by exit from zero two before his integrate. We're integrating with respect to why is Israel 24 receiver where the intersection is 24 to 4 for this function? so on. Duh. What's the function? Function is X cube things. Why, then the Why the ex? Yeah. Three are here This cape So we can integrate this sworn with respect to west Know that X cube is constant So we ever were injured. Great way were tricked Great story with respect to away which is good too Rehearse in chambers from 0 to 2. Okay, this is X cubed. I'm from around the place. But wise remember that if we have why and take off White is goingto y que y squared over too. Then we put very offer why which is good to four and zero The awful eggs. This is good too for too poor two is 16 so 16 over two is eight x cube Inject The Christian is ex cured. Yes, X cube. Listen take from 0 to 2 we put here is 16 16 over two is eight x cubed d'Or for pics. This is going to if we integrate eggs for integrates this with respected toe eggs That is X cube would be eight. Remember that we put we're happy if we have extra Poor Kay is k press one off a k press want. So it's the ex too poor for exit a pair for over four extra bar for Is that you then tour here. And one here is the ex to board next. 444 This is two X to perform two to par for is 16. You have to times X to poor form. Is it due to poor form? But he is not one into serial minus zero. Too poor for So this is 16 then The answer is 30. True. This is there. Yes, sir, for this program.

So can I Would transform the integral from x and y to you. Andy, we have two x squared minus X Y No, this was squared the ex you, Roy. And what we know is you equals two x minus y and be close to him to expose What? Okay, so here are we gonna have do teams be times that you killed in the delta X Y over will you be? And now the UDV So from here we know that the decoding for that's why I would tell you it was too the next over another you that I said Would that be? Doesn't why over data you and other way over that we The neural value for this is 1/3 So now we have well third time's the region out this g So the gsc region transformed by over x What? You'll be transformation. You know we have you become sti you devi And now we're gonna have soon from over are we know that the region is naked naked to x minus four plus four and to us for seven. And why equals two X minus two? Why customer X plus one. So here's the region for, uh, Alexis. What plea, which is called, are no interest from the region to our you'll be playing, which is called G. Let's see. So for this line, um, boy, close to four months to x. I'm sorry. Seven months to us, this goes to big was four by over relation. And for why? Because performance to X, we would transform to every week with seven for X equals y. This explains to you we transform into U equals two and the four y equals X plus one. We transform, too. U equals negative one. Okay, so right now we have all the boundary for the region. G cell week handy. Bella Too integral. We're gonna have. And that could 12 to 4 over you and the 47 for V. The inter grants are you B? We have the video respectively. In this integral will be validated too. You don't know four times you squared. Let's see from like, 1 to 2. And finally answer for this really 33 or four

A little play area going before problem number certified. The eradication of the legs can return us Linus for X equals zero playing minus for X equals two and by plus four x because right food is, like zero less than or equal quite minus 40. That's the only question group and grew less than a request by plus Forex less than a vehicles. But change off media with U equals bi minus forex, be it quits by plus for it. So are said Doctorate you Komaki given by zero last January. Corporal, you have done very questionable serial less than already cool less than a request early left and request a fight. Then the cocaine metrics the you call Marty because a date months off X comma. But so it is because while you break Alexis my nestle, you beg always what for? But it's in was, It's like minors off minus eight and was it's constantly minus like by eight. Been like a big getting by my guest for X good, But why plus for equals you, but the bigger I equals nobody did over our I made a forex dealer, but why plus for it into did you with music article? Probably a little over our you buy be. Is that called in? Off you go barfi do you being it for What do you think, Winston? The You go fight you baby one. But it David, do you? Which is equal to one? But it disease were too you. Do you? I will fight Maybe, but so really it's like one. But it youth Well, by do there are people you know. L envy Google Foot, Which is it? Goes toe want by eight intercooled Jindo Island by minus Ellard Group. What did you think was to want by four in tow? I let fired by from It's going to be late for through both fight by. Thank you.


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