5

Evaluate the integralxyz dS, where $ is that [ part c of the plane 2 =that lies the cylinder x2 +y2 = 25 using the method of your choicefirst step, set up the surfa...

Question

Evaluate the integralxyz dS, where $ is that [ part c of the plane 2 =that lies the cylinder x2 +y2 = 25 using the method of your choicefirst step, set up the surface integral for the given iunction over the given suriace $ as double integral over region R in the xy-planeSf xyz dS = Sf (Type an exact answer, using radicals as needed )

Evaluate the integral xyz dS, where $ is that [ part c of the plane 2 = that lies the cylinder x2 +y2 = 25 using the method of your choice first step, set up the surface integral for the given iunction over the given suriace $ as double integral over region R in the xy-plane Sf xyz dS = Sf (Type an exact answer, using radicals as needed )



Answers

Find the value of $ \displaystyle \iint_S x^2 y^2 z^2 \, dS $ correct to four decimal places, where $ S $ is the part of the paraboloid $ z = 3 - 2x^2 - y^2 $ that lies above the $ xy $-plane.

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.

The first thing that you have to do is figure out the bounds. So X squared plus y squared is sixteen, and our square is equal to X squared plus y squared. So this gives us that R is equal to squared of sixteen, which is four, and we're inside the cylinder. So we just have to be less than or equal to four, and our should be positive as well. And Seita not really any restrictions on theta, but we don't want to repeat ourselves. So this make they did between zero and two pi That way. No angles occur more than once, and the boundaries for Z are provided for us. Z is between four and twenty five, and then that's all their bounds. And then here we have square root of X squared plus y squared. So that's the squared of R squared. So that just gives us our So the integral becomes zero to two. Pi and Z Z, we said, was between four and twenty five, and then ah, we said was between zero and four. And then we have the square root of X squared plus y squared, which we said was R and then we have the You are FDR Easy Dee Fate which we have toe tack on to the end there. So this gives us integral from zero to two pi and a girl from four to twenty five. And this our times are gives us an r squared. So integrating that with respect are we get one third r cubed evaluated from zero up to four easy data. So that zero to two pi for twenty five, one third for cubed So four squared is sixteen sixteen times for is going to be sixty four So sixty four over three Easy the theta Okay, And now we're just integrating this constant with respect to Z. So this is just going to be that constant sixty four over three time Z Where's he has evaluated from four up to twenty five. Once we do that, we'LL have twenty five minus four here and then we have this d theta and again, this is just going to be constant with respect to theta. So once we integrate this, we have sixty for over three times twenty five minus four. So that's twenty one Time's data worth data goes from zero all the way up Tio to Pie, says his sixty four times twenty one over three times two pies. And then, of course, if you had a calculator than you could know multiplied these numbers together to get a simpler form than this. But this should be the correct answer.

Okay, let's go ahead and start this problem. We are given a three part question, but the answer is for part B and part C. I usually always do it anyways. It's basically the para militarization off the curve or the surface and the visual ization, grabbing it so that whenever you solve these kind of problems, it's just easier to see what's happening visually. In order to make sure that the dot product is taken properly, the cross product is facing the right direction, etcetera. So we're just gonna go through it exactly the same way that we have been doing it. So F is given as expert Z comma X y squared comma Z squared and the Curve C is given by Plain X plus y plus equals toe. One intersecting with the cylinder X squared plus y squared is equal tonight, so because it asks us to do the visualization, I want to go through that a little bit first. So if you imagine that Xcor plus y squared is equal to nine, is this cylinder that looks like this? We know that the radius is equal to three, and right here the center passes through the Z axis, we have a plain X plus y plus Z is equal to one and graphing that one is quite simple. If it's on the first, often it just passes through one comma, one comma, one like this. And I want you to remember that this triangular sheet, it actually extends forever. So it's more like, um, uh, she that aligns with his triangle like this. Okay, so hopefully that will help you visualize this a little bit more. So when I draw that sheet of paper so that it cuts through the cone, it will look something like this. Yeah. Yeah. And that's basically what I drew over here. Okay, so let me get rid of those Mhm. All right, Now the parametric ization. Um, using a cylindrical coordinate system would really be helpful here. So I can say that X is equal to are consigned. Fada. Why equals two r sine data Z is equal to Z, but I wanted in terms off our Insigne data. And because we know that the plane is one minus X plus y. Let's write it as one minus our times. Cosine theta plus sign data. Of course, you don't have to write it exactly the way I did. I just like Thio. Write it this way more. Okay. All right, so we're done with the primary. Parametric is ation portion and the graphing portion. So let's move on to the evaluation, Okay? All right. As usual, we are asked to evaluate the line in the group f dot d r by using Stokes there, um, we can evaluate it using the surface by taking the dog product of the curl of f with the normal. Yes. Okay. So we can calculate the curl first. Again. It's quite straightforward calculation, so I'm not gonna go through into detail what I've already done. The calculation. It turns out to be pretty nice. It's zero comma, x squared y squared. I'll put this in. Terms off are in theta later. Okay, The normal normal vector is very simple here because we are already given the plane the equation of the plane. So it is going to be parallel to one comma, one common one. So the dot product is going to be very simple here. Oh, curl of f dot n will simply be the sum of X squared and y square. So What we're going to evaluate is the double integral through s of X squared plus y squared. Oh, now I want to evaluate this double integral using thesis electrical coordinate system. So I will get double integral X is R cosine theta. Why is equal to our sign data? A Z doesn't show up, so that's good. That's good on our end X squared plus y squared does off course just simply r squared. And then we changed our coordinates from X y into our and data, So we will have r d r d feta. So the radius changes from 0 to 3. Because, as you can see, the projection off this surface onto the X Y plane is simply the circle here. Mhm. It's also given in this equation right there. Yeah, and because it's one revolution, the angle changes from 0 to 2 pi. So long story short. We're evaluating the double integral off the angle changes from 0 to 2 pi The radius changes from 0 to 3 off our cube de are de fada. So it'll simply be two pi times, um, three to the fourth power divided by four. And that simplifies to 81 pie divided by two. And that answers all of the question of this problem

I have the following, uh, surf. This will is yeah. Police off portion inside off the cylinder. Excess corpus Vice squad cause one. But it's a cylinder in the x three dimensional spaces Saluted No e unit cylinder, you know, But we'll see in a circle as the creation it goes along. Extends along. See? Well, this is, uh, under the convention that have thio the X axes seeing the wife Thetis inside of this and that along these plane along the plane given by white plus so C is equal to two. So this is a plane that intersects here. Uh, that is so It's easy. But you feel why should be too. It's it contains this point. Why is able to see your c should be one. So we designed man going through these points. Yeah. I mean, waas extensive. How? Follow over the x axis. So then the cuts should be should be a region Like what? Please, like come here like any leaps. So this is the region. The growing region. Are you want to find the area area? So for that, we need to make a final decision on the on the surface. So it's called. Yeah, leave is nowhere. I wasn't. So what would you say? That you receive all directs on business about the why so that we'll young be are constrained to be inside off the unit circle of these conditions. It's the unit circle. And then C is gonna be determined by this equation you have to see moving the why over there. Uh, this is about two. And then why the Why is he going toe this so decision Ableto tu minus V And then we realized Wait till we have sensible one minus b pops. So, uh, strategy area or the area would be the corresponding area for Yes. It is a really saw unit circle. You need circle in the plane way have this differential very at the sigma that is gonna be equal through the normal across. Programs are to view it are to be. Mm hmm. Yeah. So these, uh, can be computed us is the turning of I g cane on these better. And then you have to take its norm. So can you. Right along this row are Seville along this row rctv so out of you is that they? I think something Components with respect to you. So it's gonna be There's no tax. With respect to you would be one. That book. Why would respect to you? Zero You will see respect you zero. Okay. On the import of you would have zero or one one for why are they for Syria Would have minor Saha sort of this victor being that determine it would be equal to I comes zero. I assume so. No, I component Then, along with J Component is gonna be minus this. Components and miners have minus minus a half. So one half on that zero have been okay without to escape. Yeah, and so these vector has norm. The normal is better. He's gonna be equal toe the square off. And how squirt plus one squared. I was one of the business equal toe whole. Yeah, plus one straight cut. Did you be ableto this quote five room four that is going toe. Square it off five. Right? Like to somebody's, our differential of area has norm. So the area would be ableto uh or the region in that you will be playing received. Mom, this is equal thio for that region. This the sigma is screwed up square root of five. Upstart. You, uh, for that reason, you'll be the only plane. So it z just pull that out from visible toe school five house, and then we seem to go over. There is a few Do you re plane the leak, but this region is a unit Circle. Yeah. Circle. Well, you need this circular disk. So? So these cells, please, Part where he has his number. There is fine. I'm sorry, Squares. Or just fight. And then all the area is gonna be able to skirt of five hofs. I'm fine. Zero area off. Thank you. Division Area five over to spy, you know?


Similar Solved Questions

5 answers
Question 5 (1 point) Select the moment of the force around point D:LIO mm-200 mm125 MM200 mmF =200 N
Question 5 (1 point) Select the moment of the force around point D: LIO mm- 200 mm 125 MM 200 mm F =200 N...
1 answers
Cofactor 2X pansion auos H (uksl Iow USe to Cina 40e detemnane 0(-z 5
cofactor 2X pansion auos H (uksl Iow USe to Cina 40e detemnane 0( -z 5...
5 answers
Solve the initial value problem:(cosx)dy +ysinx= 4xcos 2 + X5 6) ~8112 , 1 Idx 36 The solution is y(x)
Solve the initial value problem: (cosx)dy +ysinx= 4xcos 2 + X5 6) ~8112 , 1 Idx 36 The solution is y(x)...
5 answers
A reicticn of 50.7 g cf Na and 56.1 of Brz yiclds 70.0 g cf NaBr. What is the perccnt = yicl? 2 Na(s) + Brz(g) 2 NaBr(s)
A reicticn of 50.7 g cf Na and 56.1 of Brz yiclds 70.0 g cf NaBr. What is the perccnt = yicl? 2 Na(s) + Brz(g) 2 NaBr(s)...
1 answers
Pio {67e Etirsti Doints Df tht (unction f(K,Y)-r> 41r' Tuan Vta Rv ?7 Derivola Test {0 jelwnnint 0nztn2 tnote poimta 06d @cml Irdrictra Jdi Iiqima- or aoudia Dointt
pio {67e Etirsti Doints Df tht (unction f(K,Y)-r> 41r' Tuan Vta Rv ?7 Derivola Test {0 jelwnnint 0nztn2 tnote poimta 06d @cml Irdrictra Jdi Iiqima- or aoudia Dointt...
5 answers
(b) A particle is vibrating according to simple harmonic motion_ The displacement vs time graph is depicted in Figure 1. Determine the amplitude of the motion. Figure 1 the frequency of the motion angular frequency the maximum speed 0 ] 8 (s)60 ms
(b) A particle is vibrating according to simple harmonic motion_ The displacement vs time graph is depicted in Figure 1. Determine the amplitude of the motion. Figure 1 the frequency of the motion angular frequency the maximum speed 0 ] 8 (s) 60 ms...
5 answers
Multiple-choice Answers: Choose the Correct Result3.55 & 2.67 3.8 2.490.52 7.33.85 & 4.8 3.493.70.63 6.47.80 & 4.26 4.8 4.490.335.39.85 & 3.11 2.8 4.490.53 4.33.85 & 3.66 4.8 3.490.53 8.5
Multiple-choice Answers: Choose the Correct Result 3.55 & 2.67 3.8 2.49 0.52 7.3 3.85 & 4.8 3.49 3.7 0.63 6.4 7.80 & 4.26 4.8 4.49 0.335.3 9.85 & 3.11 2.8 4.49 0.53 4.3 3.85 & 3.66 4.8 3.49 0.53 8.5...
5 answers
Use /Hopital's rule to find the following limit5x8 _ 6x Iim Tx3 Xzm5x3 6x lim Daype an integer or a fraction ) XIm Tx3 + 7Entet: your answer In the answer box and then click Check Answer; All parts showing
Use /Hopital's rule to find the following limit 5x8 _ 6x Iim Tx3 Xzm 5x3 6x lim Daype an integer or a fraction ) XIm Tx3 + 7 Entet: your answer In the answer box and then click Check Answer; All parts showing...
5 answers
D) Find the solution of the initial system of linear equations. The solution is of the form :=6+86 where 8 is a free parameters and 6 =sin0.0.0 0-0-C81sin (a)0-0.6 J-0-COcThe vector b may be null if the rank of A is bigger than 1. In that case you enter the null vector:
d) Find the solution of the initial system of linear equations. The solution is of the form :=6+86 where 8 is a free parameters and 6 = sin 0.0.0 0-0-C 81 sin (a) 0-0.6 J-0-C Oc The vector b may be null if the rank of A is bigger than 1. In that case you enter the null vector:...
4 answers
Use substitution and Jacobian to evaluate JS , (x+y) cos[r(x-y)ldA where D is the region bounded by the lines x+y=0,x+y=l,*-y=O,*-y=2
Use substitution and Jacobian to evaluate JS , (x+y) cos[r(x-y)ldA where D is the region bounded by the lines x+y=0,x+y=l,*-y=O,*-y=2...
5 answers
A doctor wanted to determine whether there was a relationbetween a male's age and his HDL (so-called good) cholesterol. Herandomly selected 17 of his patients and determined their HDLcholesterol. He obtained the following data. Compute the linear correlation coefficient between ageand HDL cholesterol. Round-off your answer to 3 significantfigures. Age HDL CholesterolAgeHDL Cholesterol38573844425466624634305332565136553527455240523861424955613839282647
A doctor wanted to determine whether there was a relation between a male's age and his HDL (so-called good) cholesterol. He randomly selected 17 of his patients and determined their HDL cholesterol. He obtained the following data. Compute the linear correlation coefficient between age and HDL ...
5 answers
What is occuring when an X-ray is absorbed by a metal-containingenzyme?
What is occuring when an X-ray is absorbed by a metal-containing enzyme?...
5 answers
0.2040.108CnrmanretHcunntt fllerectdettete Dobabilenc) 480} mnramun(5.2) Lunmery rmenanbanenteytnetleot Nnlne Qvotn [email protected]: NnrMuallundicnMHeloDelEileatatn[tuhax
0.204 0.108 Cnrmanret Hcunntt fllerect dettete Dobabile nc) 480} mnramun (5.2) Lunmery rme nanbanenteytnetleot Nnlne Qvotn KsJi @rahabiln Linic Fmul Imnt nulInt: Nnr Muallundicn MHelo DelEilea tatn [tuh ax...
5 answers
Arndom semple of five junior high students are 15 sclected, and each student follows: given both 4 mth und spelling test Their scoresMathSpclling1s math ability relatcd t0 spelling ability? Ieo, explain the rclationship. SHOW Your WorkII Greph your data: For the toolber_ Press ALT FIO (PC) or ALT-FN+FIO (Mnc)
Arndom semple of five junior high students are 15 sclected, and each student follows: given both 4 mth und spelling test Their scores Math Spclling 1s math ability relatcd t0 spelling ability? Ieo, explain the rclationship. SHOW Your WorkII Greph your data: For the toolber_ Press ALT FIO (PC) or A...
5 answers
1 1 1 1 1 1 1 V L1 1 1
1 1 1 1 1 1 1 V L 1 1 1...
5 answers
Maximize the objective function 4x + 4y subject t0 the constraintsX+2y $ 24 3x + 2y 2 36 Xs8 x20,Y20The maximum value of Ihe (unclion Is (Simplify your answer:)The value of x Is (Simplily your answer:)Tho ' value ofy is (Simplify your answer )
Maximize the objective function 4x + 4y subject t0 the constraints X+2y $ 24 3x + 2y 2 36 Xs8 x20,Y20 The maximum value of Ihe (unclion Is (Simplify your answer:) The value of x Is (Simplily your answer:) Tho ' value ofy is (Simplify your answer )...
2 answers
Q No. 2:(3 marks)Find power series solution of xy -(+27=-22 -2r. Q No. 3: (3 marks) Find power series solution of y=0 around ordinary point 4=
Q No. 2: (3 marks) Find power series solution of xy -(+27=-22 -2r. Q No. 3: (3 marks) Find power series solution of y=0 around ordinary point 4=...

-- 0.019882--