Use Gauss' theorem to evaluate F.ndS for the given F and o. (b) F(x, y,z) = x³i+x²yj+xyk, o : the closed surface enclosing the solid bounded by the parabolic cylinder z = 4-x2 and the planes y + z = 4, z = 0 and y = 0.

Q: Evaluate the surface integral | f(x,y,z) dS using an explicit representation of the surface. S…

A: To evaluate the surface integral using explicit  representation of the surface of function…

Q: Calculate F . dS where F = (e sin y, eº cos y, yz²) and S is the surface of the box bounded by the…

A: Here, F=<exsiny,excosy,yz2>

Q: Use Gauss' theorem to evaluate F.ndS for the given F and o. (c) F(x, y, z) = x² i+ y²j+ z² k, o :…

A: Let's find.

Q: Let S be the solid bounded by the planes z = o and y + z = 3, and the cylinder x2+y2 =2y-

A: answer is in next step

Q: Use Gauss' theorem to evaluate || F.ndS for the given F and o. (b) F(x, y, z) = x³ i+x²yj+xy k, o :…

A: (b) Fx,y,z=x3 i + x2y j+ xy k here σ is the closed surface enclosing the solid bounded by the…

Q: (a) Integrate g(x,y,z) = Inx+y over the surface of the region cut from the plane 4x+3y+2z=1 by the…

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Q: The region W lies between the spheres x2+y2+z2=1 and x2+y2+z2=9 and within the cone z=x2+y2−−−−−−√…

A: Given: The region W lies between the spheres       x2 + y2 + z2 = 1 and x2 + y2 + z2 =9 with in the…

Q: . Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = y², xy-plane and…

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Q: Evaluate the surface integral fs F. dS where F = (4x, -2z, 2y) and S is the part of the sphere x² +…

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Q: 5. Use the Divergence Theorem to evaluate [[F•d$ where F = yx² i +(xy² - 3=*)j+(x² + y² )k and S is…

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Q: Let E be the tetrahedron in space with vertices (0,0, 0), (1,0, 0), (0, 1,0), and (0,0, 1). Let F(x,…

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Q: Integrate G(x, y, z)=x4 y ( y2 + z2 ) over the surface of the cylinder y2 +z2 = 25 that lies in the…

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Q: 1 : : Evaluate F dS whereF = (r",y",) and S is the solid bounded by the ry plane and the elliptic…

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Q: (b) A solid G is bounded above by x² + y² + z² – 4z – 12 = 0 and below by z = Vx² + y². Sketch the…

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Q: 3. Evaluate if G is a solid in the first octant bounded by the plane y+ z = 2 and the surface y = 1–…

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Q: Let R denote the solid in the first octant bounded by the planes 2x + 6y + 3z = 6, x = 0, y = 0, z =…

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Q: Use Gauss' theorem to evaluate F.ndS for the given F and o. (a) F(x, y, z) = (x – z) i+ (y – x)j+(z…

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Q: Find the area of the surface of the paraboloid f(x, y) = 2 − x2 − y2 Paraboloid that lies above the…

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Q: The volume of the solid bounded by the plane 3x+6y-2z=12 and the coordinate planes is O A. 2-2y+4 1…

A: We need to find the volume of the solid bounded by the plane 3x+6y-2z=12. Now,…

Q: 13. Calculate the flux of F across S.F = (3x, xz, z2) and S is the surface bounded by the paraboloid…

A: Given vector field is  F=3x,xz,z2 we have to find the flux through F where S is the surface bounded…

Q: Integrate G(x, y, z) = y + z over the surface of the wedge in the first octant bounded by the…

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Q: Using The Divergence Theorem, evaluate F. dS where F = -xyi+ (z – 1)j + 2³k and S is the surface of…

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Q: a) Let G be the wedge in the first octant that is cut from the cylindrical solid y² + z² ≤ 4 by the…

A: To Evaluate: ∫∫∫GxyzdV. Where G: y2+z2≤4, y=2x , x=0. As per policy, we are solving only thr first…

Q: Use Stoke's theorem to evaluate the integral xydx +x*dy– 4x'ydz where C C is the boundary of the…

A: Given: ∫Cxydx+x2dy-4x3ydzBoundry square surfaces z=1vertices are (1,0,1),(1,1,1) & (0,1,1) &…

Q: Calculate F- dS where F = (3zy', ze", z) S is the surface of the solid bounded by the cylinder y? +…

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Q: Evaluate double integral f.ns where F=xi-yj+(z2-1)k and s is closed surface bounded by the planes…

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Q: 2- Use Gauss's divergence theorem to evaluate the surface integral / ry dydz + 2y°dxdz + y?zdxdy),…

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Q: Use a parameterization to find the flux do of F = z²i+ xj – 4zk in the outward direction (normal…

A: Apply Divergence theorem to the given surface integral, ∬F.ndσ=∫∫∫DivFdV

Q: a) Integrate g(x,y,z) = Inx²+y over the surface of the region cut from the plane 4x+3y+2z=1 by the…

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Q: Let F(r, y, z) = (3ry?, xe², 23 + 4.x) and I be a surface of the solid bounded by the cylinder y2+z2…

A: Partial derivative of a multivariable function with respect to a particular variable, is the…

Q: Use Gauss’ theorem to evaluate F.ndS for the given F and o. (b) F(x, y, z) = x³ i+x?yj+xy k, o : the…

A: We will use gauss divergence theorem to find value of given surface integral  If E is a  simple…

Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

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Q: 4. Evaluate //F FdS, where F(x, y, z) = (2x + 2xz, -4xy², y - z2) and R R surface of the solid…

A: The given problem is to evaluate the given surface integral over the given solid surface bounded by…

Q: A) Evaluate the triple integral ydV, E vhere E is the solid region bounded by the paraboloid – x² –…

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Q: 2- Use Gauss's divergence theorem to evaluate the surface integral || «ry°dydz +2y°dxdz + y°zdzdy),…

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Q: Evaluate f. F dS where F(x,y, z) = (4x³z, 4y³z, 3z*) and S is the surface of the solid bounded by…

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Q: Integrate g(x, y, z) = x4y(y2 + z2 ) over the portion of the cylinder y2 + z2 = 25 that lies in the…

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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,…

Q: (a) Given F(x, y, z) = –xi– y j+x²k and o is the surface of the solid bounded by the Vx² + y² and…

A: There is no specific introduction for this question. Please go through the solution.

Q: Find the centroid of the solid bounded by the parabolic cylinder 1- y? and the three planes given…

A: To find the centroid of the solid bounded by the parabolic cylinder: The region is defined as…

Q: please help me as soon as possbile

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Q: Use the Divergence Theorem to evaluate |F.n do where F(x,y,z)=(x² +cos yz, y – e" ,z² +x*) and S is…

A: According to question given :F(x,y,z)=x2+cos yz, y-ez, z2+x2S is the boundary of the solidBounded by…

Q: Use the Divergence Theorem to evaluate ∬SF⋅dS∬SF⋅dS , where…

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Q: A solid G is bounded by the paraboloid z = 4 – r - y², the cylinder 2+ y = 4 and the plane z = 5.…

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Q: Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = the planes x = 0, x =…

A: Here I am using simple tripple integration to solve this question.

Q: Let S be the piece of the surface z = 9-r that lies inside the cylinder r2 +y² = 9, and let F = (x,…

A: Given, S be the piece of the surface z=9-x2 that lies inside the cylinder x2+y2=9, and let F=x,y2,z.…

Q: Let the surface Σ be the solid bounded by the paraboloid z “ 1 ´ x2 ´ y2 and the xy´plane.…

A: Use Gauss divergence theorem to compute the outward flux.

Q: Evaluate the triple integral f te function f(z, y, :) = 5z – 3y over the salid tetrahedron bounded…

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Q: 2. The solid in the first octant bounded by the planes x = 0, y = 0, z = 1, z = 2 – y, and the…

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