5

Find the area of the surface.The part of the plane $x+2 y+3 z=1$ that lies inside the cylinder $x^{2}+y^{2}=3$...

Question

Find the area of the surface.The part of the plane $x+2 y+3 z=1$ that lies inside the cylinder $x^{2}+y^{2}=3$

Find the area of the surface. The part of the plane $x+2 y+3 z=1$ that lies inside the cylinder $x^{2}+y^{2}=3$



Answers

Find the area of the surface.

The part of the plane $ 6x + 4y + 2z = 1 $ that lies inside the cylinder $ x^2 + y^2 = 25 $

So we need to set these two guys equal to each other to find our bounds. Okay, so this is X squared. Plus Y squared is equal to three. So we're gonna use polar coordinates here, obviously, because we have some type of circle and we're going to see that are ours ranging from zero to rad three. And Artha is running from zero to two pi. Okay, so the next thing we have to do is look at our function f This is our function f We can see that F X is equal to negative two x f y is equal to negative two y. So then we just plug this into our surface area formula. Sure. I got one plus four x squared plus four y squared D A. And changing this into polar coordinates. Obviously, uh, we're gonna get one plus four r squared r d r d theta. Okay, so we see that there is absolutely no dependence on our our on data. Excuse me. So we're gonna be dealing with this integral here. Mhm. Yeah. So it actually is a U substitution. Negative. 1/4 squared of you, do you? And if we look back our bounds are going to go from one. Uh, sorry from from Sorry. Um, yeah. In a mistake here shouldn't be a negative sign from our u substitution. That's why it wasn't working. Because I have the eraser picked and it shows that 13 right? I picked you to be one plus four r squared and D u is for R d r. We're sorry. Eight rdr So this should be an eight. All right, so we've got pi over four. Evaluated three to that 13. 3 halves, minus one. Um, so we're going to end up? Yeah. Uh, Let's see here. We're going to end up with pi over six times 13 square to 13, minus one.

Yeah. In discussion we consider uh or or slender extra square less there it is square equals two war that lies about uh, scared. Yeah. Or isIS 00. 10, you know, one and one one Oh, here limits of and Yemen. Uh huh Zero is less than equals two X and X is less than equals one. Also zero is Less than equals two. Why? And why is less than equals one? No, from a ball uh, reason E is described as E equals two X, Y and zero. Yes. First then it goes to X&X is less than one. Also zero is less than it was three Y and Y is less than what? Oh, uh, surface India or a reason is equals true double in junior spread rules one glass and so divided by L X. All is bad less and said um, why? Why and why? All square? Yes. Yeah. Right. Consider the cylinder excess square less. They're square because or it can be returned as informed of that. It will do plus minus square root four minus X squared. Now we need the partial derivatives. So the partial Areva natives odd L J Yeah, I did buy Dell X is was -2 x divided by Normal people are by square root of four miles X is square. Or we can write this as and they're divided by dialects is equal to minus X divided by the square or four -X is square and I'm soon. Yeah, that's it divided by L by so it is it was zero. Yes. Uh Surface area is given by a sequel to double integral square root one plus exit square. Yeah, I didn't mind or minus X squared. Yeah. Or we can write this as double integral spread room four minus X squared plus it's a square. Why did by four -X Squared? Yeah, No later on simplifying this again we go articular by double integral one divided by spread room cough or minus X square E mm. Now on 14 the limits we get to market to buy and Giggle 0-1 Again integral 0-1 one. Do I get by square root or minus extra square Ey E X. No and continue the of all the steps. Yeah figured it was too Article about Interior 0- one. one divided what square four minus X squared E X. No, since integral one divided by this battery is square minus X. Is squared is equals two. Sign in verse X divided by hey air force you can write this as a is equal to you might be blocked by sign in reverse X divided by two. Came at 0- one. Found on applying the limits bigger is equals true sign universe one divided by Yeah. And you know that sign I Divided by six is equals one divided by two. Well weekend like this as he is equal to you sign in verse sign. Why do you get five six. But as we know that we can write this as a equals two. You I Yeah, I did buy six from other. We get is equals to buy divided by three. Therefore the surface area is Why do I did by three units?

We want to find the surface area of the part of Z equals y squared minus x squared. That lies between these two cylinders. So the formula for surface areas, the double integral of the square root of the partial square plus one. Okay, so the Z is our function F. Of X Y. Which is why squared minus X squared. So this turns into uh minus two X squared plus two. Y squared plus one D. A. Four X squared plus y squared was one D. A. Yeah. Okay. So the reason I wrote it like that, because I'm going to switch to polar now because these are cylinders and sort of be much easier to integrate. Okay, so it looks like this. So we're going all the way around the circle, 0 to 2 pi our is going from here are equals one. Out to here are equals two. We are integrating for R squared plus one R D R D. Theta. So we'll let you be for R squared plus one. The U. Equals eight R. D. R. So we need an eight and I'm 1/8. And if our equals one U. Equals four times one squared plus one which is five if r equals to the new equals four times two squared plus one 17. So this inter girl becomes 1/8 zero to pi 5 to 17 square root of you. Do you do you think them? So 180 to 2 pi you to the three halves? Over three halves which makes two thirds here. 5 17. Yeah. 1/12 0 to pi did I do that right to over 24? Yeah. Okay 17 to the three house minus five to the three house. She stated 1 12 17 to the three halves minus five to the three halves integral of juan d theta which is state of from 0 to 2 pi. So five or six times if you want 17 square to 17 minus five square root of five. Same thing as 17 to the three house. Okay.

All right. So we want Teo find the area of this surface. So remember that surface area is equal. Two double into girl over our region. Square root one plus f x square plus f y squared D a and ah, All right, so f ax. Just why f y I It's just ex. So plugging this into our double into girl square one plus x squared plus y squared. Uh, D x d. Why, however, if we look at this guy here, we can write this in terms of a circle so you can change us two polar coordinates. We see that we're just dealing with a circle of radius one. Okay, so then we see that we don't have any theatre dependent, So would you have to integrate this s o. Let's go ahead. Integrate this. So we are going to do Hey, integration with substitution. Like so Nadiya sign over here. So this is negative. Scummy plus two pi over to internal from wanted to square to you. Do you? So this is going to be high times two thirds. Two to the three over two, minus one. This is too over. Three high times two rad two minus four and wood


Similar Solved Questions

4 answers
Section 3.3 Score: 8/10 9/10 answeredQuestion 8Write an equation for the polynomial Igraphed belowytx)Question Help: QiideolL Message instructorSubmlt Questian
Section 3.3 Score: 8/10 9/10 answered Question 8 Write an equation for the polynomial Igraphed below ytx) Question Help: QiideolL Message instructor Submlt Questian...
5 answers
S17 _ 10x Write the partial fraction decomposition ofthe rational expression: ( - 3)r? + 4
S17 _ 10x Write the partial fraction decomposition ofthe rational expression: ( - 3)r? + 4...
3 answers
~/1 POINTSLARLINALG8 4.5.022.Explain why S is not a basis for R3. 5 = {(4, 1, 6), (-5,1, 3), (13, 6, 8) , (0, 5, 9)} S is linearly dependent S does not span R3 _ S is linearly dependent and does not span R3212 POINTSPREVIOUS ANSWERSLARLINALG8 4Find basis for the row space and the rank of the matrix:SG20 F3E0 Ada CFZ
~/1 POINTS LARLINALG8 4.5.022. Explain why S is not a basis for R3. 5 = {(4, 1, 6), (-5,1, 3), (13, 6, 8) , (0, 5, 9)} S is linearly dependent S does not span R3 _ S is linearly dependent and does not span R3 212 POINTS PREVIOUS ANSWERS LARLINALG8 4 Find basis for the row space and the rank of the m...
5 answers
Find the area bounded by the graphs of the indicated equations over tne given interval (when stated). Compute answers to three decimal places y=x2+3,y=2x-2 2Xr2The area, calculated to three decimal places,square units.
Find the area bounded by the graphs of the indicated equations over tne given interval (when stated). Compute answers to three decimal places y=x2+3,y=2x-2 2Xr2 The area, calculated to three decimal places, square units....
5 answers
[email protected] FuiFr < 2 hr _ u if 24* <3 i < >3f-iMntettJer4o
[email protected] Del Fu iFr < 2 hr _ u if 24* <3 i < >3 f-i Mntett Jer4o...
5 answers
Q#I: Proof the following equations without using truth table. 4) ^ (p T) and (q ^ r) are logically equivalent.4) ^ (4 - T) (p ^ 4) and (pT) is tautologyr) are nOt logically equivalent
Q#I: Proof the following equations without using truth table. 4) ^ (p T) and (q ^ r) are logically equivalent. 4) ^ (4 - T) (p ^ 4) and (p T) is tautology r) are nOt logically equivalent...
5 answers
Define f(1,y) = 12 + 3y (10 pts) Find the vector equation of the line perpendicular to the level curve of at (2, pts) Find the value of the Inaxiral directional derivative of f at (2,-1). pts) If u is unit vector which Iakes &n angle T/3 with Vf (2,-1); find the directional derivative of in the direction
Define f(1,y) = 12 + 3y (10 pts) Find the vector equation of the line perpendicular to the level curve of at (2, pts) Find the value of the Inaxiral directional derivative of f at (2,-1). pts) If u is unit vector which Iakes &n angle T/3 with Vf (2,-1); find the directional derivative of in the ...
5 answers
Solve each triangle.
Solve each triangle....
5 answers
Sketch the graph of the function using the approach presented in this section.$$f(x)=x- rac{1}{x}$$
Sketch the graph of the function using the approach presented in this section. $$f(x)=x-\frac{1}{x}$$...
5 answers
Point) Evaluate the integral by making the given substitution U2x + 3+C(2x + 3)4B. Evaluate the following definite integral. dz (2x + 3)
point) Evaluate the integral by making the given substitution U 2x + 3 +C (2x + 3)4 B. Evaluate the following definite integral. dz (2x + 3)...
5 answers
The unshaded region on the following graph reflects the feasible region based on two constraints.Which point calolies all ol the conali4nis?0 0 (7,1)(1,0)NonaUtent pana
The unshaded region on the following graph reflects the feasible region based on two constraints. Which point calolies all ol the conali4nis? 0 0 (7,1) (1,0) Nona Utent pana...
5 answers
Graph each equation.$$x+5=0$$
Graph each equation. $$ x+5=0 $$...
5 answers
9) What are the equilibrium molarities of a 100.0 ml solution that contains 40.789 mg of Iron (III) sulfate hexahydrate?
9) What are the equilibrium molarities of a 100.0 ml solution that contains 40.789 mg of Iron (III) sulfate hexahydrate?...
5 answers
Chemistr raduate #tudentgiven 250. mL ofa 0.80 M diethylamine ((CzHs)} colutian Dicthxlamina neak base with 13*10" What Mass (CzHs), NH Br chould thc student dissolve (C,HS) NH solubon bulfer Kith pH 1.62? Yau May #5umo the volume pltne solution douant chande when tne (C,4,)} NH,Br Is dissolved sure Your Jnswer has unilsymto rouno it t0 Elanincan digits:O.P
chemistr raduate #tudent given 250. mL ofa 0.80 M diethylamine ((CzHs)} colutian Dicthxlamina neak base with 13*10" What Mass (CzHs), NH Br chould thc student dissolve (C,HS) NH solubon bulfer Kith pH 1.62? Yau May #5umo the volume pltne solution douant chande when tne (C,4,)} NH,Br Is dissolve...
5 answers
Change the kondd hxtween tt cilhoon utoms cach necule ckoubl triplc txuk] #$ nccded crwupleie thc AFucMT the And shwld rcnuuin sEle bond Ien YOu d m Ied do anything Ihana Do nt = (chunec= Mher hond- the mokrule .
Change the kondd hxtween tt cilhoon utoms cach necule ckoubl triplc txuk] #$ nccded crwupleie thc AFucMT the And shwld rcnuuin sEle bond Ien YOu d m Ied do anything Ihana Do nt = (chunec= Mher hond- the mokrule ....
5 answers
The velocity of carwas read from its speedometer at 10-second intervals and recorded in the table: Use the midpoint rule to estimate the distance (in miles) - traveled by tht(s) 10/20/30/40/50/60,70,80,90/100 ulmi/h] 136/49/50/55/62/69/52/54EstimateHint 1: Take n = 5 Hint 2: Watch for the units of measurement
The velocity of carwas read from its speedometer at 10-second intervals and recorded in the table: Use the midpoint rule to estimate the distance (in miles) - traveled by th t(s) 10/20/30/40/50/60,70,80,90/100 ulmi/h] 136/49/50/55/62/69/52/54 Estimate Hint 1: Take n = 5 Hint 2: Watch for the units o...

-- 0.022051--