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Calculate $∬_{S}f(x,y,z)dS$ for the given surface and function. $x_{2}+y_{2}=16,0≤z≤4;f(x,y,z)=e_{−z}$ Consider the shown work. $T_{θ}=∂θ∂Φ =∂θ∂ (4cosθ,4sinθ,z)=⟨−4sinθ,4cosθ,0⟩T_{z}=∂z∂ (4cosθ,4sinθ,z)=⟨0,0,1⟩N(θ,z)=T_{θ}×T_{z}=∣∣ i−4sinθ0 j4cosθ0 k01 ∣∣ =(4cosθ)i+(4sinθ)j=⟨4cosθ,4sinθ,0⟩∥N(θ,z)∥=(4cosθ)_{2}+(4sinθ)_{2}+0 =16(cos_{2}θ+sin_{2}θ) =16 =4∬_{S}f(x,y,z)dS=∫_{0}∫_{0}e_{−z}dθdz $ Identify the first error in the work shown. No errors exist in the work shown. The surface integral is written incorrectly. The tangent vectors $T_{θ}$ and $T_{z}$ are incorrect. The normal vector $N(θ,z)$ is incorrect. The parametrization of the cylinder is incorrect. Calculate $∬_{S}f(x,y,z)dS$ for the given surface and function. $x_{2}+y_{2}=16,0≤z≤4;f(x,y,z)=e_{−z}$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $∫_{S}f(x,y,z)dS=$

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