10. PROBLEM Evaluate \iint_S \mathbf{V} \cdot d\mathbf{S}, where \mathbf{V} = z\mathbf{i} + x\mathbf{j} - 3y^2z\mathbf{k} and S is the surface of the cylinder x^2 + y^2 = 16 in the first octant between z = 0 and z = 5. HINT: The following FIGURE can be helpful:
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We want the flux ∬_S V · dS for V = z i + x j − 3 y^2 z k over the curved surface of the cylinder x^2 + y^2 = 16 in the first octant, 0 ≤ z ≤ 5. The cylinder radius is R = 4 and the angular coordinate θ runs from 0 to π/2. Show more…
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