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10_ Evaluate Jf(x+y )dAwhere R is the triangular region with vertices(0,0), (0,4) and (1,4)...

Question

10_ Evaluate Jf(x+y )dAwhere R is the triangular region with vertices(0,0), (0,4) and (1,4)

10_ Evaluate Jf(x+y )dAwhere R is the triangular region with vertices (0,0), (0,4) and (1,4)



Answers

$5-20$ Evaluate the surface integral.
$\iint_{S} x d S,$
$S$ is the triangular region with vertices $(1,0,0),(0,-2,0)$
and $(0,0,4)$

Mhm. Okay. This one is an integral over a triangular domain and were given the verdict is is of that domain. Okay. They are 00 11 and 40 Mm hmm. So graph it's a um Things that before 2-3 and again um not to scale. Let's put the one up here. Okay. So there and here and over here. So basically that's what the domain looks like. Okay. And uh these curves that are here that will need to set these things up. This one over here on the left side of the triangle. That's the curve X equals Y. Your wife lex. And you can work it out that this one on this side Um his ex is um 4 -3 y. Okay. You can verify that easily. You know, it's got to be a linear function and notice that affects is equal to one. Then why is 1.5 X is equal to four then? Why is zero? Okay. So anyway, this one here is what we would, you know, horizontal ease are simple, would be one term for it and we can do it in one integral if we do it this way. So what we'll do is we'll have y be the outer integral and then we'll integrate from the left side of the line, on the left to the line on the right. For each value of why? Okay. So anyway the integral were after say i Is the integral here, Why? We'll go from 0 to 1 right? And then going inside X will go from this line X is y Up to here 4 -3. Y. Okay. And uh the function they want us to integrate is just the function. Why? Okay. And so then here the X integral comes first and the Y in a girl uh comes next. Okay so that's what we have. So anyway then um when we integrate that it's going to be X. Y. So I again will be say the integral Y. Is 0 to 1. Um And we'll have X. Wine says X Had the limits of uh 4 -3 and 4 -3 y. n. y. Alright and then we'll have to integrate that over why? Okay um so again just continuing and so this is gonna be the integral Y 0 to 1. And then uh this will be um Why Times 4 -3 Y. Uh huh. Um minus Y squared right? And ultimately will have to integrate that over Dy. And so This is the integral from 0 to 1. Okay, the exes are all gone so we don't have to write that anymore. So anyway you can see this is going to be four y minus three, Y squared minus y squared. Uh So this is just um four times why minus y squared Dy? Okay. And so then this just becomes two y squared From 1 to 0 Minour wild, cubed over three from 1- zero.? And so that's just two to minus four thirds or two thirds. Okay, that's it.

Okay, here's Yeah, was set up and start with We're going toe anyway, if you want to respect the why and we're gonna go from why minus one seven minus three y for the X. So take two. And essentially we've got where you going Across this way Up and down from one. Always to in the y direction. So it's going in. Great. So just be on X y squared. So So I want to sub minus three. Why y squared minus y minus one. Why squared? See why? Yes, we want to much strain this out. Seven y squared minus three like you. Minus like cubes plus y squared. Why? Let's take it up here. That is. Want to negative for why Cubes plus eight, My squared I And like any great women, minus winds with the force over four plus eight y curable. Three want to and that's going to give us a They have 16 4 That's the heat. Times 8/3 plus 1/4 minus 8/3. See, I just more stuff girl here. My divided by Floyd er I really don't need that for it. Divided. Sorry about. And of course, it said it's not. It's no wonder here, So sorry. I think. Catch those airs for they come figure. Okay, so that's that. Fixes that. So be here. Um, let's see a negative 15. And CNN's MPs, Chevron 8/3 56. Just maybe one more step in here. Okay. Make it 40 times in a 56/3 is 11 spree.

We can rewrite the verses for this problem of 100 zero negative to zero and 004 we can rewrite these inv urgency intercept form to get X over one plus. Why over negative too. Because Izzy over four to go to one. Since these points are all on the axes from here we can multiply by four and sulfur Z go from four x minus two y plus Z equals four to get that Z equals four. Let's do I minus four x And now that we have the problem set up like this, we can use the equation that the surface integral of f of X y z MTs, is equal to the integral of f of X y g of X y multiplied by the square road of the X partial derivative squared plus the why partial derivative squared less one d A. And this would give us since we're trying to solve for the surface integral of X, this will be equal to the integral of X multiplied by the partial derivative of X, which in this case is negative four square for 16 but the partial derivative of y, which in this case is two or 41 squared plus one. Do you this me rerun as being equal to the square root of 21 on Supplied by the partial of X d. A. We now have to come up with the bounds for this function. And since we know from given earlier and equation, that's the X and Y intercept or one and two, respectively. One. And to we can draw a triangle with Vertex sees that 00 10 and zero negative too. Like so using this, we know the X over one plus why over negative too is equal to one and we can rewrite Why is being equal to two x minus two, which tells us that our bounds will be from 0 to 1 for X and for two X minus two. So why for zero? And now that we have our bounce and a formula work well, just plug in for equation himself. This will give that the square it 21 from the inter girls from 0 to 1 and two x minus 2 to 0 of x do y dianetics. This will be equal to the spirit 21 from 0 to 1 of X y from 0 to 2 x minus two two X minus two decks. This will be equal to the spirit of 21 by the end of girls from 0 to 1 of acts times two X minus two, which is equal to two x squared, minus two acts of the same integral of the same bounds. From here, we can pull out the negative from it. We'll get negative score of 21 um, of two x cute over three minus X squared from 0 to 1, and this will be equal to negative score. A 21 won't supply by 2/3 minus one. And as you go to the square root of 21/3, which represents the surface area of our integral

We realized that our s the triangular surface as part of the plane. So we have these fetuses which lie on the access. So then if we write in such intercept form of the plane, We have this then let's multiply through by four. Then if you want to apply to you by four, this is what you have. So giving that are surface is you have zero to be equal to G. Of X. Y. Then we can use this formula where there is the projection of the surface. This is a projection of the surface on the xy clean. So then so what it implies? So implies that equation of the place. So equation of the plane. We can write us zero to be equal to four plus two Y minus four eggs. So therefore this implies that Dublin seeker over the surface is eggs. The eggs is going to be the doubling in Segre over the projection D. Of eggs. You have square roots of pasha divinity with respect to X. Have minus four squared class. This is going to give us two squared bloods one the mm And this is equal to the square roots of 21. This is 16 plus 4. 20 plus 1. 21. You have the Dublin city girl? Okay. S the in. So we are interested indeed. Which is the projection on the surface of what the xy plane. But our X intercept is one since the X intercept is one. And the Y intercept is worth negative words. True from here, it's implies that our triangle. So you know you're saying X intercept is one and our white intersex intersects intercept is negative two. So this implies that our triangle has venter says zero and zero 10 And zoo. Thank you steve. Sure. So we are looking at I say this is a real sketch. This is the reserve. It's true. Let's say this is 10 Then I say zero negative two. So triangle. I think there's a triangle. We are looking at. Okay, So therefore the equation of the line joining one Saru. And so this equation of this line, It's going to be x divided by one floods. Why divided by negative two which is equal to one and can write it as Y. To be equal to 28 minutes soon. So therefore if we define the interval for the the projection. So we define the define T C. B. equals two X Y M. P. The such that the interval for eggs is from 0 to 1. And the interviewer for why? It's going to be from two x -2 20 So therefore to this implies that I think the square root of 21. Why don't we see girl over S. D. E. It's going to be called to square root of 21. The Integra 0 to 1 You have from two x -2-0 eggs. Dy the eggs. So I have square root of 21. 0-1 integrate this we have X. Y. They Ain't ever two X -20. The so you realize that the upper interval goes to zero. So we saw for lula in several and if you do you have so this is equal to negative because it's going to be minus square which of 21. So it's one you know two x squared -2 eggs the X. And this is -21. Spirit of things. What I have sued divided by three sq minus two divided by two X squared The interval from 0 to 1. And this is equal to My next question which of 21 you have So divided by three minutes 1 and this will give us Square root of 21, divided by three. So then it's in place that the integral where the surface is Of the function is the eggs. It's equal to the square root of 21, divided by three as a final and some.


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