5

12.sin(<" ) dz, where 21 =u, 2 =F0 23 [21,22,23,21] e: 13. dz. Jcsto) 2+214.fv2+2.+3)d: where is any path joining 0 to 1. 15. L(H)" 2d2. 3i 16. dz, whe...

Question

12.sin(<" ) dz, where 21 =u, 2 =F0 23 [21,22,23,21] e: 13. dz. Jcsto) 2+214.fv2+2.+3)d: where is any path joining 0 to 1. 15. L(H)" 2d2. 3i 16. dz, where ~(t) = e"t + 2,0<t < 2t [Hint: Show that the patk 2i is closed and the integrand is analytic in a region containing the path:] 17. dz, where ~(t) = i+e",0 < t < 2T. 2 +1

12. sin(<" ) dz, where 21 =u, 2 =F0 23 [21,22,23,21] e: 13. dz. Jcsto) 2+2 14. fv2+2.+3)d: where is any path joining 0 to 1. 15. L(H)" 2d2. 3i 16. dz, where ~(t) = e"t + 2,0<t < 2t [Hint: Show that the patk 2i is closed and the integrand is analytic in a region containing the path:] 17. dz, where ~(t) = i+e",0 < t < 2T. 2 +1



Answers

$15-20$ Use the given transformation to evaluate the integral.
$$\begin{array}{l}{\iint_{R}(x-3 y) d A, \text { where } R \text { is the triangular region with }} \\ {\text { vertices }(0,0),(2,1), \text { and }(1,2) ; \quad x=2 u+v, y=u+2 v}\end{array}$$

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.

Do you have me off the phone drill you saw? But you know from one 11 good point. Do you three now the miners One eyes that plus you. Dio y the X plus 61 was forced you, he said so these little can be seen as they go along. Ah, Starks. 111 on goes through through three. One on the battlefield there. Yes. So these vector field, uh, that is conservative. What's your field? Is the majority of why x poor. So, uh, it is conservative. We should be able to find some pain cell function. So that grant if Mexico do so, that means if he, uh what show or Yeah, he's, uh but is there you're drove. Were that us? Your true motives. But Blix and their movements before why their rudeness? To see along components. So that applying these differential for your toe. So by saying that this crab visible Toby, you're saying that are several have, with respect to x wife Well, the same that far. So if we respect through why, thanks. I'm that partial with respect. Well, see, photo or Children trying to solve these questions are these. Well, then yes. Should be people. Teoh. So function X times y plus function that them on. Why comes in Jim? Age have been some wild. It's easy of these all the parcels. If we respect why he's gonna people toe X, that is these first part plus the party. Why? But since this is already that way see that Bruce Parker hostile zero. So that, uh, o h this constant was a spectacle. Lights so that we have our for this equation, you know, are some affair with respect to see How far is it excess you goto x y plus inch becomes in c severe. Well, please. Well, these park Sparta's you from then. This will be here before eight students per qc extra function of a single rival on that has people before for so, uh, so that our function age she's gonna be ableto four c some constant. So we have our potential function on the basis of function is equal to ex wives. Uh, x times y plus four plus some courses. So that, um, evaluating these room that five. He's a big toe from the point. Bring point me. He's being called toe function. Never will be mine was the person that I have always had a thing on the point You still will be the function Be still and mine is one. Mine is the function of what we did at the 0.1 11 So these Ah, this is ex wife says review. I was gonna go to three. Oh, mind those four. Our council minus, uh one things one that he said Francie plus four last week. The councils I'm all of BC's five on This is six miles four. Still, it would be to minus five Somebody single, too, when it's fine. But he's, uh, miners My mystery so that these, um that is this lyinto minus, you know?

So having this problem we evaluate a double integral by changing to polar cornets. And the given integration is the double and triple over the region. R. Y squared over X squared plus y squared D. A. Where the region R. Is a region that lies between the circle X squared plus y square is equal to a square and X squared plus Y square is equal to be square. Her. A. Is less than B and A. Is greater than zero. So this is the condition we are given with. Now first I'll be sketching this area. So the shared region, this one by these lines are the is a region over which we have to evaluate the interval. The outer circle represents the equation X squared plus y square. As it calls to be square and the inner circle it rebellions. This is determinate here. This is all and inner circle represents X squared plus y square is equal to a square, notice all this. Uh Androgel, we had to change into polar cornett. And for polar corn it's you know, they're X. Is the calls to our cousin Tito why it equals to our science theater and D. A. Began right this thing us are D R D Tito. How can you use this concept here? So the double integral would become equals to the double integration. Why square would mean our square science where heroes are square, sine squared, Tito, divide by X squared plus y squared, which is equal to our square X squared plus y squared would be in our square dot. D A V F R D R D T. To if you have any confusion that how we are able to write X squared plus Y squared as our square. You can put X as here are strengthened and Y. S. R. Sorry, excess arcosanti, A and Y S. R. Santa. You'll get our square. No, this Oscar and Oscar will cancel this one. And this one next we have to put the limits for R. And theta. And here we see that the radius R. Is wearing between a baby and A. So we can say that art is ranging between A. And B. So the limit would be A to B. The 4th ERA. It is 0 to buy here. The zero and again in the coming here zero to cuba zero to buy. Now we can easily integrate this. So it will be equals two LTD 0 to 2 pi. No Science square to we are integrating with district are so Science Square theater would be concerned and our dear. So it would become our square divide by two and the liberties from A to B. And the hero. Now we can put the limits so it will be called to 0- two Pi. Science quoted as it is And this would become be square minus a square divide by two duty to. And this time we can take outside the integration. So it would become be square minus a square divide by two. Integration. 0 to 2 pi By using programmatic formula we can write that Science where Tito it is equally true. 1- Co sign. Don't try to divide by two. So this would become one minus cuisine duty to divide. But we just bring Science closer to in this form not dated them and we can take you outside this integration. So it would become for here. No again easily integrate this So it will be equals two. B squared minus a square divide by four. And here it would become tita minus the integration. Of course I intend to is signed to teeter divide where to So sign to teeter or divide by two And the limit is from 0 to 2 pi. No, simply we have to put the limits so be square minus a squared divided by four A Parliament. We have to buy minus sign two times two pi would become forward by divide by two minus the lower limits alone. But we have heard is zero And signed 0- 0- zero as it is. Well for to simplify this and get our results to be pi divided by two, be square minus a square. So this is a required answer. And finally we can conclude that the double integration oh y square over X square Bless Y square D A over the region R. Is the calls to. Why did World War Two be square minus a square? So this require answer that was asking the question. I hope you have understood the problem. Thank you.

So here in this problem we have to evaluate the given Androgel by changing into polar cornets and then google is double integration over the region day and the function is into the power X square minus Y square B. A. Where this D. Is a region bounded by the semicircle which is given by X equals two. Under look why minus sorry, four minus Y squared. This is the semicircle and the Y axis. So let me first row the region D. So this the shaded region is our region D. Right? And this Kobe is basically represented by the situation which is X equals to under four minus y squared. Now we have to evaluate this integral over this region being and to do so we have to first convert the integral in them polar carnage. And for polar carnage we can write that X as equals to arcos. In taito why it recalls to our science data D. A. Is equals to our D our duty to. So by using this concept, the integral would become so it recalls to the double integration now into the power X squared minus y squared. So this thing we can write that we can take minus common so it would become X squared plus Y squared and X squared plus vice. Where we know it is our square. So this would become negative into the power negative are square and D. A. We have our dear digital so R. D. R D. T. To or next week to see the limits No heresy that are is ranging from 0 to Here. These are zero and here it is Article 22. So the limit for our would be 0 to 2. So it would be 0-2. Next we will receive the limit for Tito Tita is wearing a ranging between five by two negative five by to hear the three days bye bye to and here the theater is negative five by so we can write that Three days wearing or is ranging from negative by by two. Too bye bye to next to it again easily. Well what does it bring along? So it would be called to the double integration. The limit would remain same. Negative five by 22 by by two. Now a to the power are square. Not to evaluate this entry goes you can't do it directly. We use the substitution method here and I'm going to substitute odd as related who I'm going to substitute are square as the so we can write that to our dear is equals two DT. All we can write our dear is equals two DT divide by two. So R D R. We have data divided by two. So DT divided by two. Now are square means T so we can write that negative are square. Would be negatively Not date heater. Now the limit for data would remain the same but the limit 40 would change because we have changed the variable here. Now if the art is zero Then T is also 0-10 square would be zero When our is to go to square would mean false. Ot would become four. Now we can evaluate this integral so it would become the double integration. Let me do this way. This is the single integration and heritage negative. Fire by two. Too. Bye bye to And integration of the power minus T. We have I left one x two here because the RDR was due to divide by two. So one x 2. I have to put it here. I'm just taken one x 2. Outside. Now if the power minus T. We have the integration would be a to the power of minus T. As it is divide by minus one to minus one. We can write it here and the liberties from 024 and data to ask this now you want to further simplify this by putting the limits so one x 2 I'm taking a serious and negative saying I'm taking outside so negative sign here and The limit is from negative by by two too bye bye to and here it would become a to the power minus four minus eight. To the power minus zero which is one not a day leader We can easily integrate this because there's a constant here so it will be equals two -1/2. And here it would be a to the power -4 -1 and the state a would become the stated that the innovation of duty to would be to to hear and the limit is from negative by by 22 by two not going to follow. Put the limits so it will be called to negative 1/2. And here it would be eating about -4 -1 and upper limit minus the lower limit. So it would be by by two- of -5 or two. So it would become plus by. But we can further simplify this and finally we'll get the results to be by a divide by two Times 1- into the bottom -4. So this is a required answer for the integral over the region. Day is the power minus x squared minus y squared E. So did the final answer. I hope you have understood the problem. Thank you.


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