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Evaluate the double integral JI5 x dA,where D is the triangular region with vertices (-3, 0), (6,0) and (0,3). [6 marks]...

Question

Evaluate the double integral JI5 x dA,where D is the triangular region with vertices (-3, 0), (6,0) and (0,3). [6 marks]

Evaluate the double integral JI5 x dA,where D is the triangular region with vertices (-3, 0), (6,0) and (0,3). [6 marks]



Answers

Evaluate the double integral.
$$\iint_{D} y^{3} d A$$
$D$ is the triangular region with vertices $(0,2),(1,1),(3,2)$

So we have this triangular region that we're going to. Graff got the origin. That one. Come one and we've got four comma zero. So here is our region. Zero zero one one. Ah, I got four zero down here and ah, we want to integrate. Ah, as region too. I believe it's two. One doesn't really matter, but pretty sure his reason to We're going to vary our exes. Okay, so we need to figure out what, this vow. What? This line is here. It's not too hard to see that. It's just why equals X Okay, We also need to figure out what this line is here. So we'LL use Ah, point slope form here. All right, so our why value is zero and what is our slope? So it's down one over. One, two, three. So negative one third x minus. R x value is for so why is equal to negative one third times X minus four. Okay, so ah, writing are integral. Our lowest value for y zero down here and our biggest one is one. Okay, now our left most line is y equals X So r inte grand again is why? And we're integrating with respects Tio ex first. And then why? So our lower bound is why And then we have to solve this guy for X. So this is negative. Three. Why equals X minus four. So X is equal to four minus three y four minus three wise. All right, so what does this equal? Well, uh, this is going to be X y were integrating this with respect to X. Why four minus three. Why? T Why? Into girl from zero to one of were planning in for X. So we got four minus three. Why? Minus y times. Why do I? So he got zero two one four. Why? Minus three y squared minus y squared D y? It's exactly what we have here. Just simplify. Not We get four. Why? Minus four. Why squared G? Why? Sorting it too? Why squared minus four over three. Why? Cubed zero two one good two Minus four thirds is two thirds

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?

So we're wanting to evaluate the surface integral, which means we can choose which formula we feel would be best given the information we have, um it appears that since we know the vergis is that we have, it will be best to use um, the formula described by the double integral of f of x y g x y times the square root of the, um, partial derivative of G with respect to X squared, plus the partial derivative of G with respect toe, why spread plus one? Yeah, we're gonna copy and paste this time to use it. In this problem, we know that our f of x y g of x y is just going to be X. And then we also know that if we take the partial derivative since we know that we can write the equation of the plane, we know to the Z equals for class two y minus four x based on what we're given, that's our plane. So we can use that to evaluate the partial derivative of G with respect to X, which is clearly going to be a negative form. And this will be a positive too someone we evaluate it. Um, we end up getting that. This is the square root of 21 which we can pull out right here. We also know that based on the bounds that were given, um, X goes from 0 to 1. And why goes from a two X minus two 20 so we can put our bounds in? We'll have 0 to 1. This will be our D Y E x or why bounds are going to be two X minus two and zero. So based on that, we get 1.5275 Um, more specifically, though, we realize that what we can dio is evaluate this. It'll be X, um, when we evaluate the integral here, x dx, we'll end up getting a 21 over three. So this all comes down to one third the way we can write. This is 21/3, and we see this is the same answer that we got. We got it before, which means that our answer has been even more so verified.

In this problem we want to write to give an integral with the order of integration interchanged without integral is the integral from 0 to 2 integral wise. Where do I. F X. Y. Z X. Y. This question is challenging understanding of multiple integral over non rectangular regions in particular how to change the order integration for such integral. So change the word of integration. We need to get our differentials and inform dy dx instead of the FBI for which we need to appropriately identify new bounds. We can do so by sketching so our region of interest is shaded on the right. Yeah, we see that are bound zero to become 0 to 4, inverting X and Y. And inverting X and Y. For functions we obtain X over two and root X. The balanchine to summarize on the right best we can write this into real ass integral. 0 to 4 integral. Route X two X over to F X. Y. Dy Dx.


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