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14. Use the Divergence Theorem to find the outward flux of F (6y)i + (3xy)j - (2z)k across the boundary of the region inside the solid cylinder x2 + y2 < 4 betwe...

Question

14. Use the Divergence Theorem to find the outward flux of F (6y)i + (3xy)j - (2z)k across the boundary of the region inside the solid cylinder x2 + y2 < 4 between the plane z = 0 and the paraboloid z = x2 + y2

14. Use the Divergence Theorem to find the outward flux of F (6y)i + (3xy)j - (2z)k across the boundary of the region inside the solid cylinder x2 + y2 < 4 between the plane z = 0 and the paraboloid z = x2 + y2



Answers

In Exercises $5-16,$ use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D .$
Cylinder and paraboloid $\mathbf{F}=y \mathbf{i}+x y \mathbf{j}-z \mathbf{k}$
$D :$ The region inside the solid cylinder $x^{2}+y^{2} \leq 4$ between the plane $z=0$ and the paraboloid $z=x^{2}+y^{2}$

All right, So we have our vector function and we have our region. So by this point where we know we're gonna use the diversions theorem to solve for this flocks through this region. So the first thing we're gonna do it's gonna be to find the divergence of So we have differential with respect to X of our first term. This is why differential with respect to why of our second term, which is X y. Then we have differential of our third term with respect Dizzy, negative Z. So this is gonna be the first term zero second term be X the third turn. Negative one. So we're just left with X minus one. So now we're gonna look at our region. We have the cylinder of X squared. Plus y squared is less than or equal to four. We have the plane as equal zero So positive values of Z up until our zeez equal to x squared plus y squared. So this little objects probably going to look a little something like this where it's a cylinder that curves out as it goes up. So when we do this, we're gonna convert everything into cylindrical coordinates so our X is gonna be equal to our radius to say the times to co sign If I why is gonna be equal to r sine phi? There's E we'll keep equal to Z now what do these angles mean? Well, if we have our three dimensional axis, we don't really need a negative disease. So this is all positive. It means that and well drawn her object here in red kind of how we pictured it beforehand. So it means that any point say on, uh right here which it's located right here at this point away, we know that if this is our X, this is or why, Mr Z, this little line segment as a distance are away from the origin and it is an angle FYI away from the X axis. And then lastly, it's left to the height of Z above our X Y plane. And those are our values that we have found here. So our differential volume is also gonna change. It's gonna go from DX dy y d c to our defy er daisy And, uh so and lastly are divergence of F, which was X minus. One is gonna change from our changed. Sorry to our co sent. If I buy this one, All right. So now we can solve. We have our flux, which is normally equal to the surface. Integral, uh, are vector field got it into our surface. But by the divergence, Dirham, that's equal to the triple integral virgins TV. So however integral when we plug in everything else our co sign with by on our devi, which you found to be far defy pr DZ. And now we solve next trick using the bounds of our region and converting them into cylindrical coordinates. So on the outside, we'll do the radius one. We scroll up and look at her original bounds. We have this circular function, which is right here. So that's just any circle with X squared Plus y squared is equal to our radius squared. So in this case, are radius squared is equal to four. So we know that the Regis of our outer circle will be to next. We'll do our angle, phi. So if I was probably around the circle so we know it's gonna be two pi And now lastly way have R Z which we know are integral is gonna be rz bounds because our cylindrical coordinates don't change anything for Z. So rz bounds air the bottom plane Z equals zero and then it's hot plane. Just gonna be the probably x squared. Most Weiss word, although however, are now in cylindrical coordinates so we can convert X squared plus y squared into our radius squared. We can do that because if we plug in our ex and why using these equations, we'll get r squared times the coastline square to five plus R squared times a sine squared If I and then using trig identities, we can factor out an r squared and see that the coastline square to five plus the sine squared If I equal to one That's why we can get our squared for our problem And now we just block everything else in So if we factor in r r r squared times consent If I minus are now we have d r in the outermost at the innermost Inderal you have defined next and then we have DZ, which I can hardly fit. But it won't be important quite yet, so that will solve our radius function first looks weight. These air reversed should be easy on the inside and the innermost integral. And then d are on the outermost Integral. Okay, now that we got that situated, we can rewrite the outer parts right now. So we have our differential with respect. Dizzy, We have nosy dependence. So we know that it's just gonna be the variable times e from zero r squared and zero We at no term. So our inner rule, our innermost integral, will just be equal to our square times Whatever is on the inside, in other words, are square tends r squared times the coastline of fine so far into the fourth Cose identify an R squared times are are to the third, Then we haven't. If I then it er so our next integral it's gonna be are defying, General. So what we do is when we take the integral of defy and we put it over an entire length of two pi, our coastline term actually goes to zero. If you think about the coastline graph and how it fluctuates in between positive and negative, the area under the curve is gonna cancel for one entire period. So when we integrate are to the fourth times co sign of by it actually goes to zero. So we don't even have to worry about that. Yeah. However, we do have our our to the third. Then there's no fire dependence. So that's just gonna be the five factored in, which is going to just end up being two pi minus zero or two pi. So we're left with two. I are cute. And lastly, we can just solve our radius integral, which is a lot easier now. And we find that we have to pie. Sorry. Negative. Two pi times are to the fourth for 4 to 0, which is equal to negative. Too high times two to the fourth or for into the fourth over four is just gonna be equal to four. So our final answer it's gonna be negative. Eight. Why

So you have before win back Tre Field. That is why on the either exam plus x y along generation, um, see my no see along the key direction I want to integrate the flux Looks out of our the region described by you have in three dimensional space, huh? The reason is bounded is between the cylinder excess for buzzwire scores Wisdom for, um is between ah c equals zero room divider. Believe. See she Uncle X squared plus y squared. So are these Looks like he's, uh, set off like followings. We have some, uh, so some reason within C equals zero. So the plane hear, exploit, that's equal zero on duh for a week service all of these, but ah, lying inside the ceiling there overuse for So we have some saying they're here. So because the vision and, uh, well, this this is Regina, that's call it see, and you see wow, like a trance. Barrasso cut should look something like that on these, Uh well, these, uh, like, rotating these two years, So you have, like, a bold So so, uh, or intimidating out, off. It's doing F, but and over disservice were amazingly no also the sea drill people Thio do any another surface off. See boundary? Yes, but in the signal. But a baby that virgins trom the physical to doing verdict. William, Did they emergence? Yes. Um, so So there is less computers from people these these days. Virgins. So these diversions people too differentiating the trust, comply with respect, works there. Why was Greg's plus? Yeah, I think with respect awhile, x y Plus, there's a minus. You'd respect Izzie, you know? So he's equal to the function zero plus x Mina's luck next month as one to like to throw over these vision inside the region. Well, danger, it'll off X minus one or C you You was a little bit first look on the angels eggs V or C so you can see there. What is what? You see these Reasoner? Well, it is, uh, symmetric with respect. Oh, the X axis. So it's not the same for every we're, uh but you see, But, uh, you see magic with respectable you respect two x so well by the symmetries. Uh, since these are not function, this interval will be zero because of your buddies. Circle around the eggs I plainly see mature, respectful They excluding it said that these internal is about zero, but that is not everything that is in there. We also have mineral of minors, uh, minors in drill, but you're outside in mine is one. So miner's angel oversee the be able. Well, this part here is able to the volume. Oh, see on so well for that, you can do the following consideration so I don't get the volume off. See on dumb Well, you see that we make a cut here. You have money in your on a Notre circle. Right? So, uh, the outer circle is equal to the ceiling there. Basile in their ex wife square people, too, for Ah. So these outer circle these eggs square was white squares. Before you say that these d x why on the sea axis from the but, you know, circle is described pointed question that sea people too. Uh, eggs square plus y squared. So I was gonna have ah is gonna be a circle overuse. Sirrah. Kel reviews Uh, um spirit of C sort of decent girl. The volume. Well, uh, we'll see you have equal to Yes. Well, the they are in this All the area off these analysts will be. Suddenly I see you let see. Go to Well, See, I was, uh I was going to hear from zero up to their wa at that point, these people to see if they would know from zero up before. Bit of zero off before off the Connelly. Well, Arya Mariel Daniel. Yeah, Very endearing. Yeah, Yeah. Finalists. That depends on C. D. C. And so from this picture you see that there The analysts has ordered a radius two. So, uh, yeah, the hubby the area off the outer circle by I can still No, I'm still squared. And then, ah, the meaner being their release us You just quoted the sea. She would be the burial that I was would be for pie for by minus Why them Cincy So So that that is eerie off not only there. So it would be the interval from zero up to four by factoring buy off Biden's for my c The sea on these This interview is about to buy times for or Z minus c squared her house. Should I be have ah well replacing within zero on four So this would be by James four squared, minus four square huffs to read 16 minus 16 house miners so that the volume of these would be Well, you won't see with people too. Ah, the damn spy. So that is our only moxie. Our flocks. You goto miners the galaxy, so we will be able to mind us eat. But so, yeah, the flux. True, these surface that to that mission. He's, uh my no say bye.

The partial derivative of this victor Expo 66 to a Asterix Roy equals 20 picks. Plus toward That's a question. Did everything with respect. Oy well, so why Plastics squares it quids to the pressure. Did everything with this pictures it well. Four Nexus Square like you equals zero before then, if equals one fix plus four plus two looks equal The triple integral over the wolf 12 x plus two y That's too do you By substituting limits of the we get triple integral from 0 to 3 and from zero Boy Where? Toe and from zero You tool Well, Twitter, our present. Tita, Let's to our saying, Peter, plus two you are is it by integrating with respect to our we get equals the government aviation from 0 to 3 from zero to buy over to Oh, me too. Peter Plus 16/3 Science eater. That's for the future, is it? Which equals the integral from 0 to 3. Well 32 less to boy plus 60/3 gives it equals 120 61 Thank you

Were given a vector field f and the surface s and were asked to use the divergence theorem to calculate the surface Integral. Oprah s of f f is extra fourth, I minus execute C squared J plus four x y squared Z K and s is the surface of the solid talent bounded by the cylinder X squared plus y squared It was one and the planes Z equals X plus two and Z equals zero first, the divergence of F is or execute plus zero plus four x y squared. And by the divergence, the're, um the surface integral over s of F is equal to the triple integral over the region e founded by S of the diversions of F, which is four x cubed plus four x y squared. And recognizing that our region E is symmetric about busy access, it makes sense to use cylindrical coordinates. Here. The projection on the X Y plane is a disk of radius one. And so we get the iterated integral, which is the integral from 0 to 2. Pi integral from R equals 01 integral from Z equals zero and Z equals X plus two, which in cylindrical coordinates is R CoSine data plus two of our function in cylindrical coordinates. This becomes for our cube cosign cube data plus four times are cubed cosine data times sine squared data. And in fact, we can combine this to get four are huge times co sign of data using triggered entities. And the differential is our Deasy DRD data taking the anti derivative with respect to Z and evaluating. We get integral from 0 to 2 pi integral from 01 of this is four are to the fourth co sign data times R cosine theta plus two. He already data and multiplying out and taking anti derivative with respect to our and evaluating you get the integral from 0 to 2 pi of let's see Siri's or yes, 4/6 or two thirds cosine squared of theta. Plus, this is 8/5 cosign data data and taking the anti derivative with respect to data and evaluating, we get well, we have two thirds times and then this will be one half times data which is two pi Yeah, which simplifies to two thirds pi


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