VI. Find the mass of a surface S, with parametrization R(u, v) = (e" sin v, e" cos v, 2e"), for Ju < 1, Jv| < n, given that the density at every point (r, y, z) in S is equal to 6z.

Q: Let S be the portion of the cylinder x² + y² = 4 between the planes z = and z = 1. Find the mass of…

A: The given shape is the portion of the cylinder x2+y2=4 between the planes z=0 and z=1. The density…

Q: 2. Let S be the portion of the cylinder x² + y² = 4 between the planes z = 0 and z = 1. Find the…

A: The given problem is to find the mass of lamina in the shape of S with the given density of portion…

Q: VIII. C ) Find the mass of the surface S, with density function 6(x, y, z) = 3r, parametrized by…

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Q: Let S be the surface defined by the vector function น V 2 2 R(u, v) , ue [0, 2], v e[− 1, 1]. Find…

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Q: Let S be the portion of the helicoid parametrized by u cos v u sin v g(u, v) = 0 <u< 1, 0 < v< 2n.…

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Q: 18. Given the vector function F=(3,-2,2yz+z) and S is the surface bounded by a paraboloid z=9-x² -…

A: The given problem is to evaluate the given surface integral of the vector field over the given…

Q: 3.Obtain the centroid of the solid formed of the area bounded in the first quadrant by y = 2x7, the…

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Q: b) Given F(r, y, z) = (r³ + cosh z) i+ (2y3 – 3r*y)j – (2² + 4y²z) k. Use Gauss's theorem to…

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Q: Answer the following True or False: Let f(x, y) be differentiable at (zo, 30). Then the vector is a…

A: The equation of the tangent plane to the surface f(x, y, z)=0 at x0, y0, z0 is defined as: fxx0, y0,…

Q: 39. Let C be the curve of intersection between the cylinder x² + y²-1 and the plane z x. Find the…

A: Given that C be the curve of intersection between the cylinder x2+y2=1 and the plane z=x.We need to…

Q: b) Given F(x, Y, z) = (x³ + cosh z) i+ (2y³ – 3x²y)j – (x² + 4y²z) k. Use Gauss's theorem to…

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Q: Let z = g(z, y) = f(3 cos(ry), y + e") provided that f(3, 7) = 4, fi(3, 7) = 2, f2(3, 7) = 3. i)…

A: Let z=gx,y=f3cosxy, y+exy f3,7=4, f13,7=2, f23,7=3

Q: Find the curvature of the vector function r(t) = (4 cost, 4 sint, t) for t = RIN 2

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Q: Calculate F . dS where F = (4x°z, 4y°z, 3z*) S is the surface of the solid bounded by the…

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Q: Let F(x, y, z) = 2xi – 2yj + z²k. Evaluate Is where S is the cylinder x2 + y2 = 9 with z E [0, 1].…

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Q: 2. Calculate the curvature, unit binormal vector and torsion for r(t) (e'cost)i+(e'sint)j +5k

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Q: 2. The vector function Ř(u, v) = (sin u cos v, sin u sin v, cos u), (u, v) E [0, 7] × [0, 2m] gives…

A: Here, the parametric vector function for the surface is given by R→u,v=sin u cos v, sin u sin v, cos…

Q: Compute the total mass of a wire bent in a quarter circle with parametric equations: x = 5 cos t, y…

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Q: Let S be the upper half of the unit sphere x +y +=1 and take n as the upper unit normal. Use Stokes'…

A: Stokes theorem is used to find the line integration

Q: 10. Use Stokes' Theorem to evaluate ||(V×F)· dS where F= (z,2x, y) and Sis the hemisphere of x' + y?…

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Q: Evaluate / . F. dS Is where F = and the surface S is the triangle with vertices (1, 1, 0), (0, 2,…

A: Given To evaluate ∬SF.dS  where F=&lt;x+1,-1-2y,z&gt; and the surface S is the triangle with the…

Q: Find the moments of inertia I, I, Io for the lamina below. Ix Iy Io II II || D is bounded by y = ex,…

A: The given Lamina is: D=y=ex,  y=0,  x=0  and  x=1ρ(x, y)=17y

Q: Calculate the surface integral // y ds, where S is the part of the helicoid defined by the S vector…

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Q: 1. Consider the spiral ramp r(u, v) = ( u cos v, u sin v, v ) with 0 < u < 3 and 0<v < 2n. Use a…

A: Consider the given information.

Q: A solid is bounded below by the cone z = √ =√x² + y² and above by the plane z = 1. Find the center…

A: The objective is to find the center of mass and moment of inertia about the z-axis if the density is…

Q: If F(x, y, z) = z - f(x, y) and s is the graph of z = f(x, y) and (a, b,c) is a point on the surface…

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Q: B. Compute the shared mass between the polar curves (r = 1+ cose) and (r = 1- cose), usę density…

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Q: Find the flux of F= through of the total surface of the closed cone y=(x2+z2)0.5 with 0 < y < 2…

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Q: Use Stokes' Theorem to find the circulation of F = (xy, yz, xz) around the boundary of the surface S…

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Q: 5. Find the mass of a plate enclosed by y = x(1-x) in the first quadrant if its density is p= Ax, y)…

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Q: 4. A surface consists of parts of the planes z = 0, z = 1, y = 0, y = 2, z = 1, in the first octant.…

A: To verify stokes theorem

Q: C. Find the center of mass of the lamina bound by y = VI, y = 0, and r = 4 if the density is given…

A: Given that, A lamina is bounded by  y=x, and y=0, and x=4ρ(x,y)=kxy

Q: x ¹yr ¹0 D is bounded by y = ex, y = 0, x = 0 and x = 1; p(x, y) = 7y

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Q: Calculate F- dS where F = (3ry², re², z³) S is the surface of the solid bounded by the cylinder y? +…

A: The general form of a vector-valued function in three dimensions is given by F¯=F1,F2,F3, where F1,…