5

5 . Calculate the surface area of the part of the plane 2 = €+y that lies between the cylinders <2 +y? = 4 and r? +y? = 9. (25 points)...

Question

5 . Calculate the surface area of the part of the plane 2 = €+y that lies between the cylinders <2 +y? = 4 and r? +y? = 9. (25 points)

5 . Calculate the surface area of the part of the plane 2 = €+y that lies between the cylinders <2 +y? = 4 and r? +y? = 9. (25 points)



Answers

$1 - 12$ Find the area of the surface.

The part of the plane $2 x + 5 y + z = 10$ that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 9$

Were given a surface integral and were asked to evaluate it. This is the integral over S of X times Z Where s is thes surface which is the boundary of the region enclosed by the cylinder y squared plus C squared equals nine and the planes X equals zero and x plus y equals five mhm. So this is kind of a complicated surface. In fact, you could break us up into three surfaces. So we have the subsurface s one which is the lateral surface of the cylinder we have surface s to which is the front formed from the plane X plus y equals five and we have s three. And this is the back which is in the plain X equals zero. So on s one the surface is given by the vector equation. Well, since this is a cylinder are of UV equals you I plus the radius of the cylinder which is three cosign v j plus three Sign the okay where V ranges from zero to pi and we know that X. We don't know much about you yet, but we know that X lies between zero and five minus why, which is the front plane. Now, rearranging this implies that you lies between still zero but then substituting five minus. And then why is in terms of U and V three cosign be so you was less than equal to five minus three co signed B Then we have that victor. Are you cross RV? Well, this is the vector one I plus you're o J plus year. Okay, crossed with zero I minus three signed the J plus three cosign. Be okay And taking this we get negative. Three cosign v j minus three sign be que and of no coefficient for I and therefore the norm of are you crest with RB is going to be the square root of nine coastline squared V plus nine sine squared V which is the square root of nine or three and therefore we have that the integral over this subsurface s one of X Z is equal to the integral bringing this as an iterated integral, integral from V equals 0 to 2 pi integral from u equals zero to u equals five minus three co sign V of our function xz in terms of U and V. So this is going to be you. Times three sign V times the norm of are you cross RV which is three. Do you TV Taking the anti derivative With respect to you, we get factory at a nine times the integral from 0 to 2 pi and then we have one half you sign the sorry one half you squared sine V from 0 to 5, minus three cosign v d B and evaluating This is nine halves times the integral from 0 to 2 pi of five minus three cosign v squared times a sign of the TV and making a new substitution in your head. Yeah, or on paper. We get nine halves times and then we have one third so one over the coefficient in front of cosign v times one over the raised exponents, which is now three. So one third time's one third or 1/9 times the expression five minus three co sign Be Now it's cubed from 0 to 2 pi and evaluating. This is five See one half of Well, actually those were identical. So this is simply going to be zero. Another way you could have figured this out is to notice that this isn't odd function. So that's for S one now on s to while recall that this is the front of our region which has prioritization r of y z equals and we have the X is given by the equation of the plane five minus Why hi Plus why j plus z k. And we have because we're constrained by the cylinder that why squared plus c squared must be less than or equal to nine. The equation of the cylinder and likewise we have that are y crossed with rz. This is simply the vector negative one I plus one j crossed with one K. This is simply I plus j and therefore the norm of our y Cross rz is the norm of this which is this squared of one plus one or route to and therefore the integral over the surface as to of X Z is what will substitute X in terms of why so this is now five minus y times z and then this is the double integral over Instead of y z such that y squared plus z squared It's less than or equal to nine and then we also have in our into grand the norm of our way across RZ which is Route two d A. And we see that this region is circular and so we might use polar coordinates here. So making the switch, we get route two times the integral from 0 to 2 pi integral from r equals zero two r equals three of and then substitution gives us five minus R cosine data times are sign data and then the differential becomes r D r D theta Simplifying This is route two times the integral from 0 to 2 pi integral from 0 to 3 of five R squared minus R cubed Co signed data Times the sign of data D R D data taking the anti derivative With respect to our this becomes route two times integral from 0 to 2 pi of now this is five thirds are cubed minus 1/4 are to the fourth cosign data from 0 to 3. I'm signed data di Fada evaluating. We get route two times integral from 0 to 2 pi and this is 45 minus 81 4th co signed data times the sign of data D data This is equal to route to times and then factoring out four for 81 times and taking anti derivatives one half of the expression 45th minus 81 4th Cosine data now squared, evaluated from 0 to 2 pi. And if we plug in zero in two pi or really subtracting same term from itself And so this is going to be zero Finally, on the last part of the surface, the back as three. Well, we have the X is equal to zero. Therefore, it follows that the integral over the surface as three of x z. This is just going to be the integral of zero, which is zero and therefore putting this all together. The integral over s of xz is equal to zero plus zero plus zero for S one s two and s three, which is zero. So our answer

For this problem, we want to find the area of the part of the plain. X plus two. Y plus three. Z equals one within the cylinder, X squared plus y squared equals three. So the first thing that we want to do here is considering that we have this cylinder that we are going to be confined to and it is a cylinder rapping about the z axis. We are going to want to parameter is the zed from that plane equation in terms of X and Y. So we can then get our sort of our of UV expression. So we would do that by subtracting X into y from both sides. Then dividing both both sides by three. Or equivalently multiplying by 1/3. So you get that Z equals one third one minus x minus two. Y. Which is the same thing as now we can write our our of UV statement as it will be you I hat Because we just said X is going to be you plus the J hat. Because we said why is going to be V plus one third? One minus u minus to b K hat? So now that we've parameter rised our surface, we want to find the U. And the partial derivatives, partial derivative with respect to you is going to be I hat plus or actually I had minus one third k hat. And then for RV it'll be j hat minus two thirds K hat. Now that we have that we want to take the cross product of our U and R. V. That's going to be determinant of the matrix formed by I J K along the top. Then we'll have 10 negative 1/3 the second row. Then we'll have 01 negative 2/3. The third row, I'm not going to go through the explicit steps of expanding as the determinant, it's pretty mechanical. Um But we should arrive at the result being one third I had plus 2/3 J hat plus k hat being our tangent vector there. So then we want to take the absolute value of our you cross fee our RV. That is because we'll be plugging that into our integral equation. So the absolute value of that just going to be the square root of one third squared plus two thirds squared plus one one square technically. And that's going to turn out to be route 14/3. So now that we have that our area is going to be the double integral, I'm just going to do a double integral over Domaine de and get into that detail of what he is in a second of just Well, it's route 14/3, the U D V. So since we have what would be the inte grand is just a constant. We can bring it out front. So we have this route 14/3 times this double integral over the domain of D U D V. Which means that actually our area is just going to be 14/3 times that the area of domain of the domain that's going to be the area empty. D. The domain is given to us by the fact that we are considering uh something with limited to be within the cylinder. X squared plus y squared equals three. Which would mean that that is going to be a cylinder with. So it's going to be an area of the circle actually in the circle with our equaling the square root of three. Since we have um expert plus y squared equals r squared is for the parametric equation. The server there. So we have our needs to look or needs to look. Excuse me. Our needs to be route three. Yeah. So we'll have then Oh actually does that check out? Let's something seems a little bit sketchy here, but we'll work through this. So we know that the area of a circle is going to be uh, pi r squared, so it's going to be pi times. Yeah, that does check out five times. Route three squared, which means that our final answer is going to be the square root of 14 high.

Here. It's natural to Emma tries in terms of, uh, said next. You know this is the advantage that two homes of Arctic of being the first and third, respectively, are 1001 on, then two more roads, especially poor differentiating with respect to the the breakables axes that, um now for why wait? We already have. Why, given in terms of side, it's, uh, 2/3 into the three IVs we retreat into print. She to get just rude, said Mama. Well, of course, that's for the dog with its not well, it is a respected. That ax ended, uh, why doesn't amount ex mortars and so that there is this partially zero now for the bounds of integration? Uh, first we could not that we're necessarily were a king, um, with positive. Why ins there? Because Dead Cube had the same sign. Dad in the square, Root said. It's only riel wins That is not negative. And, of course, then that means that Cuba's Ahnegative dissolves that square root. And, uh so why is nothing we have why and said on the necessarily, um, now to get the bounds, we can substitute 16 3rd for why castle. Oh, the 2/3 have just eight. You call too? Said to the tree house. We can square that to get 64. Is that cube taking Hue brew that to get for it. So we know that goes from zero before in order that why goes from 0 to 60 in thirds and X, of course, goes from 0 to 1. Since we're told that we're working in the the first Stockard um, no for the Jacoby in, we can see it has no I component. It does J component minus one, and kay component Rude said eso taking the horror of that factor. They have roots that squared is dead ones. Where is long? So one plus said Then you taking square rooms with square root One plus said, uh And then we can interview that easily with inspectors that just once that said that he has won't abide by 2/3 huh? On then we evaluated from said equal between zero. And that is for in the end, to go with respect tax. And the one, of course, is just one series one, Um and yes, we have to go home there. We also have to multiply this by 2/3 that the derivative of one plus sent to the three hands I canceled that 2/3 and leadership square. But now, substituting these values will have one bus for this five. The square root of five. Well, five to the three hours, which is just five room, five less one. So your final answer. Be mortified to birds. Of course, you just put one. Plus he was one. So that value whoever privates one time two birds is our aunts were this question.

Alright, women will find the surface area of the part of the plane that lies above this cylinder which I think they really should have said lies above this circle. X plus y squared equals 25 because the cylinder cuts through the plane. Okay, that's okay. That's what they meant. Surface area is the integral over the region. Square root partial with respect to X. Square plus partial with respect to Y squared plus one D. A. Okay. So before we can find the partials, this equation appear has to be solved for Z. Because it has to be written as a function of X. And Y. So to Z. This 1 -6 x -4 Y. Or Z. Which is our function of X. And y is one half minus three. X minus two. Why? Okay. And then let's let's wait on the region. We'll figure that out in a minute. Let's see what it is. We have to integrate. Okay. The derivative with respect to X is -3 Plus the derivative with respect to y -2 plus one. Okay, so that's nine plus one plus four square to 14 which is just a constant. So I'm just going to put it out here. All right now, X squared plus Y squared equals 25 is a circle with radius five. Yeah, so my choices for D. A R D X. D. Wuhai dy dx R D R D. Theta. I'm picking R D R D. Theta because that's going to be super easy because our is going from zero to r equals five. And then 3-0-2 pi. So square to 14. 0-2 pi 0 to 5. R. D. R. D. Theta square to 14 0-2 pi R squared over two from 0 to 5. Do you pay to? So that's the squared of 14 times 25/2ves times the inner girl 0 to 2 pi D. Theta. That integral is data. You plug to Pyin you get two pi score to 14 25 halves times two pi choose cancel 25 Pi squared of 14.


Similar Solved Questions

5 answers
Scp) Fovdethe major orgtnic prexluct(s) of the reactlon shown below Gxelel - Gft: ChchCOTNch00102
Scp) Fovdethe major orgtnic prexluct(s) of the reactlon shown below Gxelel - Gft: ChchCOTNch 00102...
5 answers
Given dt) = (cost) i (rint) j + tk 1:) G.) Donain L:) Grapk of r(t) c :) Unit tangent vector _ f (x,y,2) e*y2 2.) cojxy Do vf (x,9,2)
Given dt) = (cost) i (rint) j + tk 1:) G.) Donain L:) Grapk of r(t) c :) Unit tangent vector _ f (x,y,2) e*y2 2.) cojxy Do vf (x,9,2)...
5 answers
Find a, br C, and such that the cubic f(x) ax? bx2 cx + d satisfies the given conditions Relative maximum: (2, 8) Relative minimum: (4, 1) Inflection point: (3, 4.5)
Find a, br C, and such that the cubic f(x) ax? bx2 cx + d satisfies the given conditions Relative maximum: (2, 8) Relative minimum: (4, 1) Inflection point: (3, 4.5)...
5 answers
Solve the following equations. Mention the axiom(s) which justify each step in your solution_(a) 2 =-r2-4=c_7(b) x-4 =2r _ 722 _ 3r + 2 = 0
Solve the following equations. Mention the axiom(s) which justify each step in your solution_ (a) 2 =-r 2-4=c_7 (b) x-4 =2r _ 7 22 _ 3r + 2 = 0...
5 answers
If a solution of HF (K, = 6.8 x 10 has pH of 3.30 , calculate the concentration of hydrofluoric acid Express your answer using two significant figures_
If a solution of HF (K, = 6.8 x 10 has pH of 3.30 , calculate the concentration of hydrofluoric acid Express your answer using two significant figures_...
5 answers
Write an equation of the discrete-time signal shown below in terms of discrete unit step functions and/or delta functions (u[n] and/or S[n]):
Write an equation of the discrete-time signal shown below in terms of discrete unit step functions and/or delta functions (u[n] and/or S[n]):...
3 answers
Suppose jar contains 6 red marbles and 14 blue marbles If you reach in the jar and pull out 2 marbles at randomn without replacement; fird the probability that both are red.
Suppose jar contains 6 red marbles and 14 blue marbles If you reach in the jar and pull out 2 marbles at randomn without replacement; fird the probability that both are red....
5 answers
Notes:8iKLXI x 100 xi1) Relative iteration error %)2) Round all the de deer numbers out to 5 digits after the point; 3) Fill in the table below for all the snapsEjg
Notes: 8i KLXI x 100 xi 1) Relative iteration error %) 2) Round all the de deer numbers out to 5 digits after the point; 3) Fill in the table below for all the snaps Ejg...
5 answers
Find (he absolito maxuum and MInMurn, if eilhier exists, |ot I(x) =x2 Selecl Ihe correct choice below and nocossary; Illin Ihe answer boxos lo corriplole your choice,Tno; absolute ITIINIMUM [$ alx= 0 B There I$ no absolule minuumSelect Iho correct choice below and, necessary; Iill ir1 Iho answor boxos (o complete vour choice 0 A; The ahsolute maximum (5 alX= 0 B. Thero IS no absolule maximumCllck to select and enter your answer(s).
Find (he absolito maxuum and MInMurn, if eilhier exists, |ot I(x) =x2 Selecl Ihe correct choice below and nocossary; Illin Ihe answer boxos lo corriplole your choice, Tno; absolute ITIINIMUM [$ alx= 0 B There I$ no absolule minuum Select Iho correct choice below and, necessary; Iill ir1 Iho answor b...
1 answers
Use the graph of $f$ to sketch each graph. To print an enlarged copy of the graph. (a) $y=f(x)+2$ (b) $y=-f(x)$ (c) $y=f(x-2)$ (d) $y=f(x+3)$ (c) $y=2 f(x)$ (f) $y=f(-x)$ (g) $\quad y=f\left(\frac{1}{2} x\right)$
Use the graph of $f$ to sketch each graph. To print an enlarged copy of the graph. (a) $y=f(x)+2$ (b) $y=-f(x)$ (c) $y=f(x-2)$ (d) $y=f(x+3)$ (c) $y=2 f(x)$ (f) $y=f(-x)$ (g) $\quad y=f\left(\frac{1}{2} x\right)$...
5 answers
1. Find the Laplace transform of the given function defined on the interval t 2 0 by evalu- ating the integral F(s) = Jox f(t)e-stdt.(a) f(t) =t (b) f(t) = te-t (c) f(t) =t2
1. Find the Laplace transform of the given function defined on the interval t 2 0 by evalu- ating the integral F(s) = Jox f(t)e-stdt. (a) f(t) =t (b) f(t) = te-t (c) f(t) =t2...
5 answers
Below are some fatty acids commonly found in foods. Classifyeach fatty acid as saturated, monounsaturated, orpolyunsaturated.Refer to the diagrams of fatty acids in foods in question 1.Coconut oil is about 48% lauric acid, 16% myristic acid, and 10%palmitic acid. All of these acids are [ Select ] ["saturated","monounsaturated", "polyunsaturated"] and therefore we expect that coconut oilwill be a
Below are some fatty acids commonly found in foods. Classify each fatty acid as saturated, monounsaturated, or polyunsaturated. Refer to the diagrams of fatty acids in foods in question 1. Coconut oil is about 48% lauric acid, 16% myristic acid, and 10% palmitic acid. All of these acids are ...
5 answers
Question 2 (a) Provide a full mechanism for the following reaction which explains the likely ' major product formed. (assume the most stable enolate is formed): marks]NaOH(b) Using ' the table of acidities provided, calculate the equilibrium position for the step involving formation of an enolate in this reaction, and suggest why NaOH is a suitable choice of base in this case_ 3 marks] [Total Question 2 = 7 marks]
Question 2 (a) Provide a full mechanism for the following reaction which explains the likely ' major product formed. (assume the most stable enolate is formed): marks] NaOH (b) Using ' the table of acidities provided, calculate the equilibrium position for the step involving formation of a...
5 answers
On a planet far far away from earth /, IQ ofthe ruling species is normally distributed with a mean of 103 and a standard deviation of 16. Suppose one individual is randomly chosen: Let X-IQ of an individualFind the Inter Quartile Range (IQR) for IQ scores_01IQR
On a planet far far away from earth /, IQ ofthe ruling species is normally distributed with a mean of 103 and a standard deviation of 16. Suppose one individual is randomly chosen: Let X-IQ of an individual Find the Inter Quartile Range (IQR) for IQ scores_ 01 IQR...
5 answers
Prove for all n € Z+-2 31 += - (n+1)! (n+1)!
Prove for all n € Z+- 2 31 + = - (n+1)! (n+1)!...
5 answers
What type of inheritance pattern is this? How do you knowWhat are the genotypes of the parents?
What type of inheritance pattern is this? How do you know What are the genotypes of the parents?...
5 answers
A spherical shell ([=2/3 MR?) rolls with constant velocity and without sliding along level ground. Its rotation kinetic energy is: Half (1/2) of its translational kinetic energy B. two-fifth (2/5) of its translational kinetic energy two-third (2/3) of its translational kinetic energy D. double (2/1) of its translational kinetic energy Same (1/1) as its translational kinetic energy
A spherical shell ([=2/3 MR?) rolls with constant velocity and without sliding along level ground. Its rotation kinetic energy is: Half (1/2) of its translational kinetic energy B. two-fifth (2/5) of its translational kinetic energy two-third (2/3) of its translational kinetic energy D. double (2/1)...
5 answers
The correct affirmation for the integral is=30 J (1_x)i.)Ifu = 2x, the output integral isJ (2 _ u) " du ii.) Ifu=x -1,the output integral is= 30 02 (1+u30) . du iii:) Two out of 3 integrals are correct iv.) Ifu = x/2: 30 32 33 Jo (1 _ 2u)" 2 dlu v:) AIl 3 integrals are correct
The correct affirmation for the integral is= 30 J (1_x) i.) Ifu = 2x, the output integral is J (2 _ u) " du ii.) Ifu=x -1,the output integral is= 30 02 (1+u30) . du iii:) Two out of 3 integrals are correct iv.) Ifu = x/2: 30 32 33 Jo (1 _ 2u)" 2 dlu v:) AIl 3 integrals are correct...

-- 0.019885--