5

Is the hexagon in the cy-plane with vertices (5, 5) , (5,5) , (0,10), (-5,5) , 5,-5) , and (0,~10) , directed counter-clockwise. 3y 3c Evaluate dx dy: 22 + y2 22 + ...

Question

Is the hexagon in the cy-plane with vertices (5, 5) , (5,5) , (0,10), (-5,5) , 5,-5) , and (0,~10) , directed counter-clockwise. 3y 3c Evaluate dx dy: 22 + y2 22 + y2

is the hexagon in the cy-plane with vertices (5, 5) , (5,5) , (0,10), (-5,5) , 5,-5) , and (0,~10) , directed counter-clockwise. 3y 3c Evaluate dx dy: 22 + y2 22 + y2



Answers

$D$ is the triangular region with vertices $(0,0),(2,1),(0,3) ;$ $\rho(x, y)=x+y$

All right, here we go. We're gonna integrate. We have this region here in the origin. This point over here 12 and this point of here, which is 03 We were triangle that comes in here. Make. So So the boundary we're gonna integrate inside here and are integral. Looks like this. And we have to decide according to this boundary, how we're gonna do this. And I've decided that Oh, I'll set it up like this. So my direction here will be this vertical direction. They don't integrate along the bottom on that strip. So I'm gonna go and here set this up right inside. Looks Why y the X otherwise going to run from the bottom? And if you look a little bit of work here, uh, you need the equation of the line. It runs through these two points. You can see the slope is too and runs to the origin. So that's fairly easy to see that that's just wife was two x this one of here. Maybe not so easy to see it, but, uh, it has soap of negative one, and it cuts through a 03 So it must be why equals negative. Next, Most three. Okay, so those are the two values for why we're gonna go from bottom up to the top, right? There's three minus X enforce excludes from left to right. And we're going from cereal toe one, which encompasses an entire, um, left to right area. So those in the laundries? Toronto one. Okay, so then your four school and integrates, Um, the first integration would make that a That's why square it's two with a two. We're going from wyffels two x toe. Why? Equals we're going to Why those two x two? Why cords three minus sex. He And until another step here, we gonna end up with Yeah, all the tools will drop out, and we'll just end up with three minus x squared X and n minus a four x cute. Okay, um, for a little more work to here, I'll expand with nine minus six x plus X squared times that X and then that's going to be I next minus six x squared, and we'll have a pause after the third minus four x of the third. So one more step before we start in here nine x minus six X squared finest Marie X to the third. There's about two X. Okay, Um, Let's see. Well, look. Looks okay. Um Okay, so, uh, it's good, cause there. So you were down here, So we're going to have a, um integrating nine x squared over two. No more room there, minus that six. Expert third over the three. Minus reaction of floors over four value away from sealed one. Okay. And then that becomes that. 0 to 1. We will end up with a Lets You have 9/2 minus to minus a 3/4 which becomes every well, I can't do a little bit more. 18 04 minus 8/4 minus rial four. So that makes 704 OK?

I really get Question 27 which gives us some coordinates. So we're gonna start by just sketching a quick graft to get a feel for what we're looking at. This is the technique that I use and our first corner is negative 7 10 So that would be appear somewhere that will call that why? And I'm gonna make a note that it's negative seven Common 10. And then our second coordinate is, if positive 5 10 So we'll say that's right here. And that's our Z, and we need to find P. And so what we want to do with that is look at the other things they told us. We know it's a 30 60 90 right triangle, and he is in the fourth quadrant, which means it's gonna be down here somewhere in Quadrant four and it tells us the right angle, Izzy. So in other words, I'm gonna have this straight line here and then this straight line down here and somewhere down here is where P is going to end up, but we don't know where exactly yet. Um, so if this is a 30 60 90 right triangle, I can count how long this is. And so since it's negative seven and positive five, that's a total distance of 12. Which means that I then know that the high pot news has to be twice as long because this is the short leg, and then the high pat news is gonna be twice as long. So that would be 24. I also know that for P, my ex value is going to match up with Z because that's my right angle. So since he has an X value of five, my P is going to have an X value of five. So I know I'm gonna have the point as my final answer five something. But I need to figure out what that something is so I can go to my question and I'm gonna call this point why? And I know that the distance between why and P is 24 so I can use my distance formula and say that 24 which is my distance, is gonna equal the square root of five. So my ex one minus my ex two. So five minus negative. Seven. All squared plus y minus 10 All squared. So now we're looking at just solving this with some algebra. So I like to get rid of the square root first. So I'm gonna square this and I'm gonna square. This 24 squared is 5 76 I don't have to worry about the square root anymore here because I squared it so this cancels out with that. So I have five minus negative seven. That's well squared is 1 44 Plus, I have y minus 10 squared. And I always tell my students, when you see something like this squared, write it out for what it is, because otherwise you'll square it incorrectly for most students. So in other words, right out y minus 10 times Y minus 10. That way, you don't lose the middle term, since I can tell from the Y minus 10 times Y minus 10. It's gonna be a quadratic. I'm going to subtract the 5 76 from both sides so that I have zero on this side and that's going to give me on my calculator here Negative for 32 and then I'm going toe multiply out. So I have why squared minus 10. Why minus 10 wine. Close 100. So also did was double distribute or foil or whatever method you learn. So I did. Why times? Why was why squared White Times Negative 10 was the negative 10 y and then the negative 10 times each of the parts. So now I can combine like terms, so zero equals. Why squared? Minus 20 y and then it's going to be minus 3 32 so that, um, I can solve it now looking at that, I'm guessing it's not going to solve nicely, so it's not going to be something that I can factor. Um, part of the reason I'm guessing that is cause 3 32 looks like a pretty crazy number to try and get it. Factors that multiply that would add to be 20. And then the other reason is because the the directions told us in simplest radical form, so most likely it's not going to factor nicely. So I'm going to jump to my quadratic formula because that'll work. Whether it would affect it or not. A is equal to one. B is equal to negative 20. If you can just pull these from the equation and plug him in, that's fine. If you can't then write him on the side like I'm doing. And then I'm using usually quadratic form that we say X equals. But since my variable is why I'm making it, Y equals will have negative be. So that's negative. Negative. 20 plus or minus the square root of B squared minus four A. C A is one and C is the negative. 3 32 my mounds all over. Two a. One or two times one. So now we're just gonna do some algebra to solve this. So negative negative is positive. So that gives us 20 plus or minus two squared a negative 20 squared is 400. Then I have negative four times. One which is negative four times negative. 3 32 which means that's like adding And that turns out to be 13 28 Gibbs, and that's all over, too. So now why equals 20 plus or minus the square root of 17 28? I checked that on my calculator. It doesn't have a perfect square, so I simplify the radical. So when I take the square to 17 28 and I looked at its factors, I end up with 5 76 goes into it, which is the square to 24 um, and then square to three. So this is equal to 24. So it's 24 square to three. So when I simplify this Wow, no happen there. Um, was to raise that part for you. So why equals? 20 Divided by two is 10 plus or minus. The 24 divided by two is 12 square root three. Now we have two answers. Here we have 10 plus 12 square to three, which is gonna be a positive number. But since our why is in quadrant four, we don't want a positive number. We actually want a negative number. And if we do 10 minus 12 square root of three, that will give us that negative number. So our final answer here, I was going to be 10 minus 12 square root of three for our y value. And we're finally done

In this problem we wish to evaluate the given integral. The double integral in region de of like you D. A. Where region D. Is bounded by or rather a triangle inside burgess 0 to 11 in three to this question is challenging understanding of how to evaluate and integrated integral to do so. We utilize single variable integration technique. Step by step and step one. We evaluate are an integral and step to report the result of our inner integral into the integrated outer integral which we saw. So determine what our bounds of immigrant or rather our limits of integration are. As well as the orders are differentials. Let's draw our region. Our region is bounded here. It is highlighted. Now we see that we have to non horizontal lines. X equals two minus Y, Y equals two minus X and x equals two minus one to minus one or y equals one half plus one half X bound in our region. Thus to integrate what we're going to do is going to integrate an X. From the left line to the right line. And then integrating why? From 1-2? Yes we have integral. One to integral to minus Y tu tu minus two Y minus one Y q b x D Y. The internet integral evaluates as three Y one S three. That's another integral is integral. 12231 of the fourth minus three QB life. This evaluates these three wide of the fifth over five minus three Y to the fourth over four. From 1 to 2 which gives final solution 1 47/20.

Right here on this problem. We've got a triple integral to intervene before we actually look at that. Let's just look at getting our bounds right and just getting our bounds right on the now we wanted underneath X Plus two y Lizzie's, too. And so that's the same Assane Z is equal to minus airs, minus two. Why? And so for my first integral here for my DZ part, I know Zito's from zero. The two minus X minus two y starts at zero because we were told that it is also bound by the coordinate planes on your coordinate planes. That would be when Z is zero. All right, so there's rz part now for the rest of it. Let's come back up into the original Make Z zero. And so this tells us we have X plus two. Y equals two. It's only from down here a graph that X plus two wise to. So I'm just grabbing this in the X y plane. That's why make Z zero. It's why Menzies zero sorry. Plus two, Why equals two And so it tells me I have this line here a y intercept a one x intercept of to And then some sense bound by x axis and the y axis. It means we're just looking at this train your region here. And so let me start with, uh, let me do acts. X goes from the Y, axis over to the curve from the Y axis over to the curb. So the curve in terms of X would be to minus two. Why, Yeah, that's what that Linus so X goes from zero two to minus two. Why is your 02 to minus two? Why? So for my next integral here, I need to make sure that I LaBella's DX it goes from zero two to minus two. Why, right? And then, lastly, to find our bounds on why part here to find our bounds on? Why, Why? I went from the X axis to one and why has a maximum value of one smallest value of zero. He would start on the X axis. That's a wise interval. Rose from zero. So one that outside interval there should always be just Constance, you shouldn't have any dependent. Inter inte grams there on that outside with it should always be just constancy. Clean this up in the morning and come back up there and actually evaluate this. And that was just getting our bounds here. And the actual inside of this interval were told this four y z So we have four. Why? Z, I just got take this piece by piece. Here. Let's start with that. Is the d Z part? Yeah. Now the anti driven of four Z With respect to Z, you had one of the exponents divide by the new exponents. That's too Y z squared. Why? Just constant. So it just stays. Why? And then we evaluate this from why equals? I'm sorry. From Z equals zero. You know more from Z equals zero that bound up top, which is two minus X minus two. Why? Z equals zero to minus X minus two. Why? And so this becomes too. Why? Times tu minus X minus two. Why? I swear. No, we may want to understand this all out. I know this is gonna be a little ugly here, but when we expand this all out is going to be really early. I tell you what, we don't need to explain. Now let's just leave like this. It was use u substitution on the next, ruinously like this. So now we got rid of the Z Park. Now we still have the integral with respect to X, which is 02 to minus two y and it goes from zero Tu minus two. Why? I still have a Y on the outside of the same idea. So the why on the outside we're just taking it one interrelated time, though, so let's just take care of this. And so the first thing that you need to do let's take the anti derivative of two. Why? Times two minus X minus two y to the second with respect to X imagery and take it with respect to X. And when we do this, this is going to give us two. Why remember to always just constantly two. Why Times X plus two y minus two to the third over. Three. They just add one of that exponent divide by the new exponents and then divide everything by negative one because undoing the chain rule here. And so this goes from X equals zero X equals two minus two. Why? Hey, because those were our bounds here. And so you're doing a plug in zero, plunging to minus two. Why? And then subtract those disappointing zero francs. Plug in to minus two y for X. You'll notice you get a lot of stuff. Cancel out here. When we do that, this is going to give us negative 16 third's. Why times why minus one to the third A party into my eyes to why for X start subtracted and it gives us this. We saw that after Interval that at her intervals the interval from 0 to 1? Yes. And now, once we get here, we still need to take the integral from 0 to 1. With respect to why so, first again, you need to take the anti derivative of what is inside there. When we take the anti derivative, it is an ugly mess. Your best bet is to foil out why minus one to the third. And then it's just your industry power rule all the way across here, right? I'm gonna save the messy algebra for you. You're showing you how to do you triple intervals. Now the anti derivative, whenever you foiled out and then use your power Will is negative. 16 15th wind of the fifth was four wide of the fourth, minus 16 3rd wide of the third, plus 8/3. Why squared? That is our anti drifter there. And then we won't evaluate it from y equals zero toe one. When you evaluated a zero, it's just gonna all go away because this is just phone home. So basically all you're doing is just plugging and one everywhere, then seeing what it gives us. So it would be negative 16/15 times, one plus four times one minus 16 3rd times one plus 8/3 times one. That's what we're going to add these together, and that should give us 4/15. So that is our triple interval 4 15


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