U SIII u' cos u?, EXAMPLE 6.6.3 For a surface of revolution defined by x u! oin o? COS 3 flol) aub omo f dn of alago C2 fnd the Gaussign curuature K.

Q: Let C be the curve y 6x° for 0 < a < 1.9. 40+ 36- 32 28- 24 - 20- 16- 12- 8- 0.5 1.5 Find the…

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Q: Find ervstion for the tangen plare to the Surface an Siven by x2y?3 =12 at the point Z-2,1,3)

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Q: 13 - x² – y? at (3, 2, 0), then If the point (1, 1, k) lies on the tangent plane to the paraboloid z…

A: z=13-x2-y2 Partially differentiating with respect to x, we get zx=0-2x-0=-2x Partially…

Q: 1. Use the Stoke's Theorem to evaluate y'dx + xy dy + xz dz, C where C is the bounding curve of the…

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Q: 1. Let F(x, y, z) = yi+ xj+ yzk and the surface S be the part of the paraboloid z = 4 – x2 – y? that…

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Q: QUESTION 8 Find the area of the cylindrical surface in Xyz-space where a point (X,y,z) on the…

A: Find the surface area of the cylinder in xyz-space where a point x, y, z on the surface satisfies…

Q: (b) A = 2pza, + 3z sin ø a, – 4p cos p a̟ and S is the surface of the wedge 0 0 < ¢ < 45°, 0 <z < 5

A: Let us solve the b part step by step with given A value

Q: 28. F(x, y, z) = x³i+y°j+z°k, and S is the sphere of radius 3 and centered at the origin.

A: divF→=ddxx3+ddyy3+ddzz3        =3x2+3y2+3z2        =3x2+y2+z2  by divergence theorem,…

Q: 9.3.16. Compute the surface area of the surface obtained by revolving the given curve about the…

A: (a) To compute the surface area of the surface obtained by revolving the given curve about the…

Q: Which of the following is the surface area of the surface generated by revolving the curve y = in…

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Q: 3.use Greens theorem S(1+x dx + 2xy dy ,where C is triangle With vertices (0,0),(1,0).(1,3) oriented…

A: "Since, you have asked multiple questions, we'll solve the first question for you. If you want any…

Q: Find the area of the surface generated by eY + e -y revolving the curve x = in the interval -In 2…

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Q: 1 Find the area of the surface obtained by rotating the curve y = In(1 – x²), 0 < x <. about the

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Q: Evaluate the surface integral I F. dS when F(x, y, 2) = y²i+ 3x² j and S is the graph of - ža (1+æ²…

A: Here, Fx,y,z=y2i+3x2jz=121+x2-y2

Q: ix of F across the surface defined by R(u, v) = (2u, 3v E [0, 1].

A: We have to find the flux according to the given information

Q: 17. Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1x 4, 0 y < 1

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Q: 3. Evaluate , (x + z)dS where the surface S is the first-octant portion of the cylinder y² + z² = 9…

A: Step:- 1 Since, we know Surface integrals ∫∫Sf(x,y,z)dS=∫∫Df(x,θ) rx×rθ dxdθ…

Q: 1. Evaluate SSSE z dV where E lies between the spheres x2 + y? + 2² = 1 and a? + y² + z² = 4 in the…

A: Given, ∫∫∫EzdV where E lies between the spheres x2+y2+z2=1 and x2+y2+z2=4 in the first octant.

Q: Jse Green's theorem to evaluate the ne integral (2tan x _cos 2y³) dx + (7⁰s y + 2x (πº where is the…

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Q: The surface area of the surface generated by revolving y=1-x? for 0<x<1 about the x- axis is None s=…

A: surface area S =∫012πy.ds where ds=1+(dy/dx)2

Q: Q1 (9 M)/ Use Gauss's theorem to evaluate ff. F.ds where F= 4xi - 2y'j +z°k and S is the surface…

A: We have to evaluate the integral ∬SF·ds where F=4xi-2y2j+z2k and S is the surface bounding the…

Q: Question 12 Evalvate the to $2xy dx + (x+y) dy C is the OF y=o direction line integral by applying…

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Q: Given a surface 3xy - In (4z – 3) = 0, where z = decimal places. f(x,y). Find zx at y = -6, z = 1.…

A: We have given  3xy - ln(4z - 3) = 0             3xy= ln(4z-3)             4z-3= e3xy…

Q: Evaluate ∫c (y+sin x) dx + (z2 + cos y ) dy +x3 dz where C is the curve r(t) = (sin t, cos t, sin…

A: ∫C y+sin x dx + z2 + cos y dy + x3 dz = ∫C y+sin x i + z2 + cos y j + x3 k (dx i + dy j + dz k)…

Q: The area of the surface obtained by rotating the curve y = 1 I+ X, 1 <xse about the line x = 28 is…

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Q: Question 4 The surface integral 3x + 6y do V1 + ely + 422 over E= {(r, y, z) E R³ | 0<I<1, r -1 <yS…

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Q: Suppose F(x, y, z) = (x + yz, 3 cos(xz) – 7y, 5z + eª). Suppose S is the ellipsoid y? 22 4 9. 16 k…

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Q: (d) Let 0 arg(z). Take C to be the wavy contour in the z-plane described by 0 < arg(z) < T; |2| =1-…

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Q: Which of the following is the surface area of the surface generated by revolving the curve y = 폴 in…

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Q: .... ........ 4.8. Find the area of the surface generated by revolving the curve x = cost, y =…

A: See the details solution in below

Q: Given a surface 2xy - In (5z - 2) = -10, where z = f(x,y). Find Zx at y = -10, z = 1. Leave your…

A: Partial differentiation

Q: Find of the ay+2²7 the revolving Curve x = 2 0≤y in 2 about the y-axis is The area, S, of the…

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Q: The surface area of the solid of revolution generated by rotating 3 x = (y² + 2)ž,1 sys2 about the…

A: Given function is x=13y2+332 where 1≤y≤2. The surface area of revolution of the function is given by…

Q: The volume of the solid generated by the curve y = e 2, between z = 2 and z = 6 about the a - aris…

A: Solution-

Q: If S c R³ is the unit sphere and F Sls curl F n dS. (e*y, sin(22 + y), xyz), then evaluate the…

A: We will find out the required value.

Q: 1 I. Evaluate dV, where E lies between the spheres x? + y² + z² = 1 and x2 + y? + z² E x2 + y? + z?…

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Q: Use either form of Green's Theorem to evaluate the following line integral. 03x y dx+ 4x y dy; C is…

A: According to the given information, it is required to find the value of integral using Green’s…

Q: Given a surface 3xy - In (3z - 2) = -5, where z = f(x,y). Find zx at y = 3, z = 1. Leave your answer…

A: This question is based on Partial differentiation of function

Q: Given a surface 5xy - In (2z - 5) = -1, where z = f(x,y). Find zx at %3D = -9, z = 1. Leave your…

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Q: Oue Sol ue have Surface ズ= 4パーナ at (-11214) DHerentiak partially fimidız) with r to 12 Sx= 811 Nou…

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Q: Let R be the region inside the ellipse x = 4 cos 0, y = 3 sin and outside the circle x = 2 cos 0, y…

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Q: The surface area of the surface generated by revolving y 1-x? for 0Sr<1 about the x-axis is S=…

A: Rule- If the curve y=f(x), a≤x≤b is rotated about the x−axis, then the surface area is given by

Q: Which of the following represents the are a of the surface of the portion of the cylinde 2+ z? = 4…

A: We need to find the integral which represents the area of the given surface. This can be done as…

Q: 1 (#3) Find the surface area for y = 3 on [0, 3] rotating on the x- axis.

A: Given an equation of a curve, The solid will form by rotating the curve about the x-axis.

Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

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Q: if Vx 1 dA if R is the unit circular disk. Find Lütfen birini seçin: O A. 3r(22-1) O B. 27(/2-1) OC…

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Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A: First we have to find partial differentiation with respect to x , y and z of function f such that-

Q: Q#4 a. Find the area of the surface generated by revolving abuut the x – uxis the portion of the…

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Q: 10. Sketch the following surfaces (graphs of functions) for a = 2, b = 1: (a) Elliptic paraboloid.…

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