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Compule the (lux the vector lield downward orienation_(xy, 4y2 S2*| through the portion the plane 3x 2y + 2 =first octant with the...

Question

Compule the (lux the vector lield downward orienation_(xy, 4y2 S2*| through the portion the plane 3x 2y + 2 =first octant with the

Compule the (lux the vector lield downward orienation_ (xy, 4y2 S2*| through the portion the plane 3x 2y + 2 = first octant with the



Answers

Use Stokes' Theorem to evaluate $ \iint_S \text{curl} \textbf{F} \cdot d\textbf{S} $.

$ \textbf{F}(x, y, z) = x^2 \sin z \, \textbf{i} + y^2 \, \textbf{j} + xy \, \textbf{k} $,
$ S $ is the part of the paraboloid $ z = 1 - x^2 - y^2 $ that lies above the $ xy $-plane, oriented upward.

This question are the circular region enclosed boy X squared plus y squared equals four. So the reintegration off regions sigma off f got and yes, equals double integration off this region off to Y square minus one DD, which is equal. Double invigoration, off toe order square, Sinus square seat a minus one and our limit zero to Zito to buy and are the hors d zita. So these will be equally for boy.

And this question We look at one of the bombs so we haven't see equals two x squared plus y squared. So this tells us that we should use the cylindrical coordinate system. So let us just put this at the top right hand corner for easy reference. Next, we want Thio convert everything into cylindrical form. So X squared plus y squared equals R squared. So this would be one of our bonds and then our other bound Z equals 23 y. We can substitute y for our sign data, so that would be our other limit for Z. Um, we can also look at a combination on thes two. So when we equate them together, we have a bound for our. So our would be by default. I would start from zero because our reverse toe radio so it cannot be negative, and then the upper limit would be three sign data and we know that Z iss Ida um, the bones for Z is three y and x squared plus y squared. And, um so we know that Z would definitely So from these goes to express plus wife's where we know that no matter what value X and Y take Z would definitely be positive. And by extension, if Z is necessarily positive why, it must be positive as well. So from this we have our limits for data. So it would be between zero and pie. We can Then we write our integral as such. So the function we want to integrate um iss are square and we feel in our limits. So we first integrated with respect to see So as you can see, the function does not continue to see. So essentially we are integrating. See, with breast out We are integrating one with friends to see So we have Z we feel in the upper limit and the limit expanded and we have three r to the power of four sign data minus R to the power of five which we can then integrate with respect to our So, um we would and then we fill in the upper limit and lower limit for our in this case, Take note that the upper limit for our is three sign data and after filling it in, we have some coefficient and sign data took the power off six. So let's look at some data to the past six first, because we cannot integrated directly. So you want to try to expand it on manipulated Such that we can, um, integrate each time directly. So we first, um, rewrite sign data to the power of six as signed Data Square. Whole thing to the power off. Three. So science data squared can be rewritten. ISS half times one minus school saying to data. So that's how you get your first step. And then in the second step, E, um, we take out the coefficient. So after the power off three, we take it out, put it in front, and then we expand, um, one minor school sign to say the whole thing. Cute. We expand it. And then there's a cool sine squared term and a co sign cube term. So those terms, we would not be able to integrate them directly. So what we want to do is to walk on them photo. So for coastline to Data Square, we can rewrite it as half times one plus co signed fourth data and then, as far consigned to Data Cube, we can rewrite it as consigned to data times, consigned to data square and consigned to Data Square. We can rewrite it. Ask half times one plus tickle sign for data. So in the next step, which is in the fourth line, we have consigned to data co sign for data co sign to data and our last term still looks a little complicated. We cannot integrate co sign for Arteta. Times Co sign to data directly. So what we wanna do is to use a trigonometry go identity to separate them. So we have, um, co sign six data plastic assigned to Eva. So just be a little careful about the manipulation here. We should have one or develop 1/8 times five divided by two minus 15/4 times. Consigned to data plus three three divided by two times. Co sign for data minus a quarter. Times CO signed six data. If you did not fact arise 18 out. You should be able to check signed data to the power of six should be equal to this. Um, And since we have isolated sign data to the power of six, we might as well um, do the integration right here. So, um, way can integrate. It turned my term. We integrate five over to respect to data, we get five forward to data 15 divided by four times. Consigned to data becomes 15. Divided by eight. Sign to data. So because coastline to data because sign to data times half and, um three over to co sign for data becomes 3/8. Signed fourth data 1/4 coastline six data becomes 1/24 times. Sign six data. We feel in the upper limits and lower limits. And we get 243 pie divided by 32 in case you're wondering where 243 divided by 10 comes from, Um, right before the integral it came from where we, um last left off before we looked into science data to the politics.

In the problem, this is the curve and we have to evaluate they dig real. That is in the problem, as we have to find the girl F dot ds. So as we know that formula double integral carl f dot ds equals single integration f dot. They are so we are going to evaluate this. So after evaluating the integration, we obtained the value of integration, F. becomes zero and this is the answer.

So left off the have super means. Why has the last word With zero we can choose to variable, you know, freeway. So actually who's using was beat and we solve y for the Parametric representation of y the one minus you square by three fi square over to take a square root because we want y to be that non positives. So we put a negative sign here.


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