(1 point) Use the divergence theorem to calculate the flux of the vector field \( \vec{F}(x, y, z) = x^2 \vec{i} + y^2 \vec{j} + z^3 \vec{k} \) out of the closed, outward-oriented surface S bounding the solid \( x^2 + y^2 \le 4, 0 \le z \le 4 \). \( \iint_S \vec{F} \cdot d\vec{A} = 160\pi \)
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The divergence of a vector field F = <P, Q, R> is given by div(F) = ∂P/∂x + ∂Q/∂y + ∂R/∂z. In this case, F = <0, y, 0>, so the divergence is div(F) = ∂(0)/∂x + ∂(y)/∂y + ∂(0)/∂z = 0 + 1 + 0 = 1. Show more…
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