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Points) Evaluate the integral with respect to surface area12x dA where T is the pant. of the plane x +y+4z = 9 in the first octant:1 12x dA...

Question

Points) Evaluate the integral with respect to surface area12x dA where T is the pant. of the plane x +y+4z = 9 in the first octant:1 12x dA

points) Evaluate the integral with respect to surface area 12x dA where T is the pant. of the plane x +y+4z = 9 in the first octant: 1 12x dA



Answers

Use double integration to find the area of the plane region enclosed by the given curves. $$ y^{2}=9-x \text { and } y^{2}=9-9 x $$

So for this problem are going to be one. He's in the formula, that thes surface integral of f x y z the US is equal to the integral of f of our UV quantified by the magnitude of the cross product of you envy what do you deviant? And so we were given precision that why squared plus Z squared is equal one. This will be equal to a cylinder, um, with an X axis at the axis with radius of one. So if you want a parametric eyes this, um, this will give us the co sign you assigned you and since s is the part of the cylinder is in the first auctions X and Y both have to be positive. That's only going to happen when you was between zero and pi over. So it's just us circle. And since V we can see right here, a V is equal to acts and we know that our bounds both ex being equals zero and X equal to three tells us that the bounds for V will be zero and three. Now that we have all that information set down, we could start working towards solving the problem. First thing we need to do is come up with the magnitude of the cross product of unity. To do this will need both EU partial and V partial, but says R is equal to V I plus co sign you of J plus sign You okay? The u partial are you will be equal to zero. I minus. Sign you, Jay. Let's co sign you. Okay. And then the V a partial will be equal to I plus zero j plus zero k as neither of the J or K components have a bee in them. So I only cross these two together. Are you cross RV? This will be equal. Teoh I j ok of zero negative sign you co sign you of 100 and this will be equal to I times Negative sign you a co sign you up saying you 00 minus did a shay component zero assigned you 10 plus k time zero negative sign you 10 and then solving this will give that the cross bonnets of the U. N. V Partial is equal to zero. I plus co sign you, Jay. Let's signed you okay. And so the magnitude of the cross product be able to the square roots, the cross product squared. So is your escort plus co sine squared plus sine squared in Khowst and script of science crew is equal to one. So the square root of one we'll give that the main suit across parts is equal. One. It's now plugging into our ah formula from earlier. Uh, we know that the surface integral of Z plus X squared Y t s. We can use her parents ization and substitute out z x and y In this case that will give out sign U plus v squared because excess square multiplied by co sign of you and I will be almost supplied by one you d. V Now from here. Once you plug in, we'll have that. This is equal. We'll have that. This is equal 2030 to pi over two. A sign U plus v squared co sign you. Do you devi And this is integral Weaken solve without having to do any substitution or similar tricks to solve this problem. And this will give ounce that this is equal to the angle from 03 of V squared plus one DV, which is equal to the cubes over three plus V from 0 to 3, and that will be equal to three cubes over three and plus three. Three cubed is 27 27 or three is nine so nine plus three until it is that the service area were integral is 12.

So given the surface s, um and we know the equation Z equals G of X Y. What we're gonna do is use this formula right here so we can calculate the surface in a room. F of X y z ds is going to be equal to the double integral over the region of F X Y g of X y times the, uh, square root of the partial derivative G with respect to X squared, plus the partial derivative with respect to y squared plus one d A. So with that, we write the equation of the plane that were given as e equals four minus two x minus two y Onda. With that, we take the partial derivatives and what we end up having is that this is going to be equal to, um, the double integral of X z d s, which is then going to be equal to the double integral over the d region of X times for minus two x minus two y times the integral of negative two squared plus negative two squared plus one d A. So that's just the integral or three times the integral over the D region of four x minus two X squared minus two x Y d a. Um And since that is just part of the plane in the first oct int, we are going to rewrite the equation of the plane and intercept form. So that's gonna be X over two plus y over to plus Z over four Equalling one. And we see that X and Y intercepts are both too. So the D region is going to be a triangle. A right triangle with Vertex sees 00 to 00 tomb. So with all this information, we know that X is between zero and two and wise between zero and two minus X Then we can now evaluate this integral. The bounds of integration will be from 0 to 202 to minus X with a three out in front. And then what we have is four x minus two X squared, minus two x y d Y d x We evaluate the first in a room and what we end up getting is three times the interval from 0 to 2 of yeah X times two minus X squared DX and then depending on whatever property use, you could use substitution method or something like that. And what we end up getting as a result is going to be that this is equal to four. So that will be our final answer for this surface integral.

Were given that the bounds of X or from 0 to 3 and the bounds of wire from 0 to 2 since he has given to us in terms of X and why we can solve for the surface integral by B formula the f of x y z ds is equal Teoh integral of f of x y g of x y times the square roots of the partial derivative of X squared, plus the partial derivative of y squared plus one d a. So now we just need to come up with those values, so that would be for F of x y glx y you will be have the integral of X squared y times one plus two x plus three Why times the square roots of the exper Shal, which is going to be too so two squared, plus the white partial, which will be three. So three squared plus one d A. Um, solving this further will give that thescore it's of 14 times the material of X squared y plus two x cubed. Why plus three x squared y squared d A. Now to evaluate the bounds were given that the rectangle goes from 03 by 02 So that tells us from before that X goes from 0 to 3 and why it goes from 0 to 2. So far, Integral will have the skirt of 14 from 03 and 0 to 2 of X squared y was two x cubes. Why plus three X squared y squared D Y D X. This is equal to the square. It's of 14 from 0 to 3 of X squared. Why squared over two plus X cubed? Why squared plus X squared y cute from 0 to 2 of DX. From here, we're just going to plug in the values from 0 to 2 and then evaluate the X one. That's forgive that this is equal. Discourage 14 from 0 to 3 of two X squared plus four x cubed plus paid X squared DX, and this is equal Teoh skirt 14 from 0 to 3 of 10 X squared plus four x cute DX. Now to evaluate the ex central, this will be equal to you. Square at 14 of 10 X cubes over three plus X to the fourth from 0 to 3. Solving this will give 171 times the square root of 14 and this will be the answer of our surface integral

If the soffits. So as surface S. Is given by excusing bye Z. To be equal to the funding stream G of X. Y. Then we can use the formula. So you have the W. And z gra over the surface is you have your function X. Y. Said The S to be equal to the Dublin Segre over the self esteem to have F. The function X. Y. Then said is G. Of X. Y. Then the square roots of so you have the pressure. They'll be waiting for G. With respect to x suede plus the pasha. They get evolve G. With respect. So why squared like this one? The A. Where we are saying that our X. Y. Belongs to. Mhm. So we are the is the projection of the surface on the X. Y. Plane. So projection B. J. Shane on the X. Y. Clean. That is the so giving that as as part of the plane. So giving giving that our A's as part of the plane of the plane implies that said it's equal to one plus 2. X plus three. Y. So this implies that you have the Dublin Segre. Uber. The surface is which is equal soon. This is which is X. Squared why? Z. D. Eggs. And this is going to be equal to. We have the Dublin Segre over D. We have X squared Y. We have one plus two. X. Plus three way. Which is Z. In the square roots of two square plus the derivative of G. With respect to X. S. Two plus. That's of why is 3-plus 1 D. A. So this is equal to the Dublin city girl. Over the surface. D. You have X. Squared white spread then one plus two X plus three. White. This is four plus nine. Which is it's a scene Plus one which is 14. So let me right. Four plus nine plus one D. A. So this gives us square roots of 14. As circumstance out. You have X squared. Why? If you expand plus two sq y plus three x squared y Q D. A. So evaluate this so the surface lies above the rectangle. So we are told that the interval for that And several to three then 0- two. So as to this implies that our aces within this interview to then why is also within these things ever. So then I don't win. See girl here X squared Y. Z. The ace Is going to be square roots or 14. You have from syria to theory Then 0- two. We have X squared Y plus two S cube Y plus three X square y que wake you weigh squid. So this is why squared Y squared here. The way the eggs evaluate this So and those who give us 14 square roots of 14 evaluate with respect to y. 1st Do you have from 0-3? And we have two X squared last four S cube plus it's X squared the X. And this is this will be equal to square roots of forward scene. Yeah. You and see you guys there were 23. You have sane X squared same X squared Plus four. Ace Cube the X. And this is equal to square roots of 14. I have saying SQ divided by three plus eggs to the powerful X. To the power for the interval from 0-3. So you realize that your lower interval goes to zero. So you have square roots of 14. You have sane three Cube divided by three bloods, 3 to the power for. And this it's equal soon. 171 Then square roots both 14, which then it's a finer answer. So then you conclude by saying this. Did the wind see Girl Uber The surface is of the function X squared. Why save the S. It's equal to Square roots of 40 117 one.


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