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18. Find JI F*ds where S is the part of the paraboloid 2 = 4 -x2 _y2 which lies above the square 0 < xs1, 0 Sys | with upward orientation. The vector field is F=...

Question

18. Find JI F*ds where S is the part of the paraboloid 2 = 4 -x2 _y2 which lies above the square 0 < xs1, 0 Sys | with upward orientation. The vector field is F=(xy,Yz,zx)

18. Find JI F*ds where S is the part of the paraboloid 2 = 4 -x2 _y2 which lies above the square 0 < xs1, 0 Sys | with upward orientation. The vector field is F=(xy,Yz,zx)



Answers

In Exercises $19-28,$ use a parametrization to find the flux $\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma$ across the surface in the specified direction. $\mathbf{F}=4 x \mathbf{i}+4 y \mathbf{j}+2 \mathbf{k}$ through the surface cut from the bottom of the paraboloid $z=x^{2}+y^{2}$ by the plane $z=1$ in the direction away from the z-axis

For giving an integral. We were asked to evaluate Simple and the role of X plus y plus C off the region K, where he is the solid in the first, often that lies under the tabloids. Ecause four minus X squared minus Weisberg While this parable Lloyd Z equals four minus X squared minus y squared, this intersect the X Y plane, which is the plane Z Po zero in a circle X squared plus y squared equals four or influential coordinates. This is R squared equals four, and then a Zara's positive. This implies articles, too, and so and so into a coordinates. Our region E If you set of triples our data Z, it's the fatal eyes. Well, because when the first often realize between zero empire but to are is going to lie between zero and two and Z will lie between zero The X Y plane and between the Paraiba Lloyd four minus X squared minus y squared. Reaching rectangular, rectangular or mixed. Switched to yeah, political coordinates is four minus R squared. And so the triple Integral um, experts y plus z over the region E. This is the iterated integral, which is integral from 80 to pi over two and drop Marco 02 in the Balkans equals zero for minus R squared of a function in terms of cylindrical coordinates. So this is our cosine theta plus our sign data plus Z 20 differential, Which for cylindrical coordinates, this is our times Easy BRD data within the anti derivative with respect to Z we get and they grow from zero to pi over to you go from 0 to 2 one back going out on our hold Arkan's ours are squared co sign data plus sign data eyes mm plus one half times are times Z squared from Z equals zero To see those four minus r squared DRD data with the anti derivative Sorry. Evaluating integral from zero to pi over two integral from zero to and this is or are squared minus art of the fourth Times Co sign the A plus sign data plus one half times are times or minus r squared, squared DRD theater and taking the anti derivative with respect to our this is integral from zero to pi Over two of here this is four thirds are cute minus 1/5 part of the fifth times Co. Sign data plus sign data and then this next term. Do the use institution in your head We get, see one half times negative one half times one third This is negative 1 12 times the inter function or minus r squared to the new power three from r equals 02 Data and evaluating you get integral from zero to pi over two and then plugging in. This is 64 15th times the cosine of data plus the sign of data and then plus and 12 times for cute, which is 16 thirds deep. Data taking anti derivative with respective data. This is 64 15th times, man man, today of your fear is going to be signed data minus cosign data. A 16 3rd data data equals zero to pi over two. Evaluating you get 64 over 15 times one minus zero plus 60 16 3rd times pi over two minus 64 15th times zero minus one minus 63rd time. Zero because zero and we simplifies to two times 64 15th 1 28 15th plus a thirds pi

All right. So you want to go ahead and solve the Portugal already of X plus Y plus E. T. V. Yeah but it is just uh the region in the first act in yeah, bounded by the probable. Oid Z is equal to four minus X squared minus Y squared. Mhm. Now we know that the first accident Gives us two bounds on our payroll. So we know that it means that Z. is greater than zero and that data must be between Pi over two and 0. And to find your balance and arm we can just said um are tabloid equation which in cylindrical coordinates is for minus R squared Equal to zero. Since we know that's are found. That's the x. Um the bottom surface that we're talking about. So this just gives us R squared equal four or articles to. So then our last um val dishes that ours between zero and two. So you can go ahead and convert this entire thing. The cylindrical coordinates knowing that devious shifts are easy t already data. So this just ends up giving us It's a girl from 0 to Pi I'm sorry not to brian pi over two. It's a girl from it's here to to to go from 0 to 4 minus R. Squared of our coastline data. Which is X. In cylindrical coordinates R. Plus are assigned data. Which is why. And let's see our dizzy. Do you already paid up? Yeah let's go ahead and drag this interview down here and we can go ahead and multiply it out with ST bounds giving us our square times co sign data. Scientific data plus a Harzi T Z. D. Already data. Right? So the first thing you can do is integrate with respect to Z. Okay. Mhm. Which should give us since this does not have a Z. And this does we can just go ahead and plug in our bounds immediately. So we know that this term will just live. Leave us with um Z in front and then this term we'll just leave us with as you squared over two. So if we go ahead and plug in our bounds we'll see that. Um This first term becomes R squared co signed data for science data. Mhm. Times four minus R squared and then zero gives us zero. So you have to worry about that. And then the second term gives us our times four minus R squared squared all over two. And then again 00 So we don't have to worry about that. Do you already data? Mhm. And then we can go ahead and simplify this further. We can go ahead and multiply both out. So leaving us with are you sorry for R squared minus hearts? The fourth could decide data. Side data. Okay. Plus and then we're gonna go ahead and square this in it. Let's apply in the yard at the same time. So we have All over two. So this gives us um 16. We'll go ahead and do the R. After. So we have 16 -8 R Squared Plus. Hard to the 4th. And then if you go ahead and put the the So the R. N. We have 16. Our last eight are acute. It's hard to the fifth. The arctic data. So now we can just integrate respect to our giving us the following. So we have 4/3 R cubed minus hearts to fit over five times. Co science data. Science data. Oh and then we can do this on our hands. So this this term becomes a it R. Which will give us for R squared. This term becomes a four hour cube which gives us minus are to the fourth and this term gives us Artists 6/10 or 12 from Jared to do data. Now we can go ahead and plug in our bounds. We'll see that zero gives us zero for all the ours. So we can go ahead and just see that this is four times R cubed. So 24 times 8/3 32/3 -32/5. Because I'm fada was signed data plus We have 16 -16 plus 64/12. You pay them This cancels 64/12. I just equal to 16 over um three. So then we have It's a girl from 0 to Pi retune 32/30 minus 32 or five and go ahead and do that on the side. So we have 32 times 1 3rd 25th Which is equal to 32 times a five -3/15 Or 64/15. So we have 64/15. Co science data science data plus 8/3 are sorry 16/3 you data. And then we can go ahead and integrate this. Finally giving us 64 15 scientific data minus coastline data Plus 16/3. They don't From 0 to Pi over two. Okay. And then we can go ahead and evaluate this 64 or 15 times will separate both. So this is the left parts. We have sine of pi over two which is one minus cosine of power to zero and then subtracting sign of zero which is zero. And then subtracting my minus cosign theta, tau and Kazan zeros once we have this and then plus 16/3 times prior to This is just one plus one or two. So we have 128 over 15 to us eight thirds five as our final answer

This problem, we want to find a parametric representation of the part of the hyper polaroid. Four X squared minus four, Y squared minus x squared equals four that lies in front of the wise airplane. Now we can do this bye. Setting the x and Y components of our surface to just being X and Y. Or we could say if we wanted U. And V. To make it more clear that those are being set as parameters. And then we can rearrange that equation that were given to isolate zed in terms of U and V. So we can do that pretty easily. We would subtract four from both sides and then add Z squared to both sides. So we have four X squared minus four, Y squared minus four equals Z squared. Which then means that said is going to equal plus or minus the square root of all of that. We could do a little bit of simplification here. Not super necessary in this case. Uh The other part is that it does say it lies in front of the wise, it plain significance. There is the fact that we do have two routes here plus or minus. But because we're taking the part that lies in front of the wise that plane, we take the positive route so to keep things consistent in the way that we represent this can write this as you I hat plus the j hat, what's b J hat plus the square root of four times you squared. Remember we parameter Rised acts as being you minus four V squared minus four K hat.

For this problem, we wanted to find a parametric representation for the part of the sphere, X squared plus y squared plus Z squared equals four that lies above the cone Z equals the square root of x squared plus y squared. So to begin, we can parameter rise uh this surface by saying just okay, well X is just gonna, let's say that's going to equal you and let's say that, why is going to equal now? We can rearrange the given equation to get zero in terms of X and Y. Which we can then get they get the Z coordinate in terms of U. And V. So we would have that said squared is going to equal four minus X squared minus y squared. Which then means that Zed is going to equal plus or minus the square root of four minus x squared minus y squared. Now we do need to be careful here because we are told that we are looking for the part that lies above the cone Z equals the square root of x squared and y squared. So one moment here. The last part we do need to address is the fact that we are lying in the region above this cones that equals square root of x squared plus y squared. So essentially that is giving first of all, that tells us that we are going to be taking our positive route there and then in addition for that requirements to be satisfied, essentially, we need to have that said is going to be greater than um not greater than or equal to zero needs to be greater than X squared plus y squared. Which then by extension means that the square root of four minus x squared minus y squared has to be greater than the square root of x squared plus y squared. We can reduce this down eventually. We should get that's X squared plus Y squared needs to be less than or equal to two for this to be satisfied. So we have our coordinates for our parametric representation, as well as our sort of boundaries on it.


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