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Problem 2. Evaluate (a²z+y²2) ds, where S is the top half of the sphere x² + y² +2²=4. 167

Problem 2. Evaluate (a²z+y²2) ds, where S is the top half of the sphere x² + y² +2²=4. 167

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Can please solve problem #12 and show all of your work in pictures do not type it, please! make sure you do the right work, thank you for your help.

[[ (a²z+ y²2) dS, where S is the top half of the sphere S x² + y² + z² = 4. 16 Problem 2. Evaluate ][ (z+x²y) ds, where S is the part of the cylinder S y² + ² = 1 that lies between the planes x = 0 and x = 3 in the first octant. 12 Problem 3. Evaluate Problem 4. Evaluate that lies in the first octant. 4 Problem 5. Evaluate • [[ 20 S that lies between the planes z = 1 and 2 = 3. 364√/2π/3 J xz dS, where S is the part of the plane 2x + 2y +z = 4 Problem 6. Evaluate ]] x ds, where S is the triangular region with vertices S (1,0,0), (0, -2, 0), and (0,0,4). √21/3 Problem 7. Evaluate ]] xz dS, where S is the boundary of the region enclosed by the cylinder y² + ² = 9 and the planes x = 0 and x + y = 5. 0 !! dS, where S is the paraboloid y = 9-x² - 2² S together with its cap when y = 0. (7/6)[37/37 1] +9T Problem 8. Evaluate. Problem 9. Evaluate. x²² ds, where S is the part of the cone z² = x² + y² //* F. ds, where F(x, y, z) = xyî + 4x²ĵ + yzk, and S is S the surface with parameterization F(u, v) = ui + vj + ueºk, where 0 ≤u≤ 2 and 0 ≤v≤1. 16(1-e) Problem 10. Evaluate · // ₁ F. ds, where F(x, y, z) = (x, y, z²) and S is the S part of the unit sphere where x > 0, y > 0, and z < 0. 57/24 Problem 11. Evaluate [] F. ds, where F(x, y, z) = (y, z-y, x) and S is the surface S of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (0,0,1). -1/6 Problem 12. One important law of electrostatics is Gauss' Law, which says that the net charge enclosed by a closed surface S is Q = €0 E ds, where •!! B €0 is the constant permittivity of free space. Use Gauss' Law to find the charge enclosed by the cube with vertices (±1, ±1, ±1) if the electric field is E = xi+yj + zk. 24€0
Transcribed Image Text:[[ (a²z+ y²2) dS, where S is the top half of the sphere S x² + y² + z² = 4. 16 Problem 2. Evaluate ][ (z+x²y) ds, where S is the part of the cylinder S y² + ² = 1 that lies between the planes x = 0 and x = 3 in the first octant. 12 Problem 3. Evaluate Problem 4. Evaluate that lies in the first octant. 4 Problem 5. Evaluate • [[ 20 S that lies between the planes z = 1 and 2 = 3. 364√/2π/3 J xz dS, where S is the part of the plane 2x + 2y +z = 4 Problem 6. Evaluate ]] x ds, where S is the triangular region with vertices S (1,0,0), (0, -2, 0), and (0,0,4). √21/3 Problem 7. Evaluate ]] xz dS, where S is the boundary of the region enclosed by the cylinder y² + ² = 9 and the planes x = 0 and x + y = 5. 0 !! dS, where S is the paraboloid y = 9-x² - 2² S together with its cap when y = 0. (7/6)[37/37 1] +9T Problem 8. Evaluate. Problem 9. Evaluate. x²² ds, where S is the part of the cone z² = x² + y² //* F. ds, where F(x, y, z) = xyî + 4x²ĵ + yzk, and S is S the surface with parameterization F(u, v) = ui + vj + ueºk, where 0 ≤u≤ 2 and 0 ≤v≤1. 16(1-e) Problem 10. Evaluate · // ₁ F. ds, where F(x, y, z) = (x, y, z²) and S is the S part of the unit sphere where x > 0, y > 0, and z < 0. 57/24 Problem 11. Evaluate [] F. ds, where F(x, y, z) = (y, z-y, x) and S is the surface S of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (0,0,1). -1/6 Problem 12. One important law of electrostatics is Gauss' Law, which says that the net charge enclosed by a closed surface S is Q = €0 E ds, where •!! B €0 is the constant permittivity of free space. Use Gauss' Law to find the charge enclosed by the cube with vertices (±1, ±1, ±1) if the electric field is E = xi+yj + zk. 24€0
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