(Solved): Calculate Sf(x,y,z)dS for the given surface and function. x2+y2=16,0z4;f(x,y,z)=ez C ...

Calculate ${\iint }_{S}f(x,y,z)dS$ for the given surface and function. $x^2+y^2=16,0\le z\le 4;f(x,y,z)=e^{-z}$ Consider the shown work. $\begin{array}{l}{\mathbf{T}}_{\theta }=\frac{\partial \Phi }{\partial \theta }=\frac{\partial }{\partial \theta }(4\cos \theta ,4\sin \theta ,z)=⟨-4\sin \theta ,4\cos \theta ,0⟩\\ {\mathbf{T}}_z=\frac{\partial }{\partial z}(4\cos \theta ,4\sin \theta ,z)=⟨0,0,1⟩\\ \mathbf{N}(\theta ,z)={\mathbf{T}}_{\theta }\times {\mathbf{T}}_z=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ -4\sin \theta & 4\cos \theta & 0\\ 0& 0& 1\end{array}\right|=(4\cos \theta )\mathbf{i}+(4\sin \theta )\mathbf{j}=⟨4\cos \theta ,4\sin \theta ,0⟩\\ \Vert \mathbf{N}(\theta ,z)\Vert =\sqrt{(4\cos \theta {)}^2+(4\sin \theta {)}^2+0}=\sqrt{16\left({\cos }^2\theta +{\sin }^2\theta \right)}=\sqrt{16}=4\\ {\iint }_{S}f(x,y,z)dS={\int }_0^{2\pi }{\int }_0^4{e}^{-z}d\theta dz\end{array}$ Identify the first error in the work shown. No errors exist in the work shown. The surface integral is written incorrectly. The tangent vectors ${\mathbf{T}}_{\theta }$ and ${\mathbf{T}}_z$ are incorrect. The normal vector $\mathbf{N}(\theta ,z)$ is incorrect. The parametrization of the cylinder is incorrect. Calculate ${\iint }_{S}f(x,y,z)dS$ for the given surface and function. $x^2+y^2=16,0\le z\le 4;f(x,y,z)=e^{-z}$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) ${\int }_{S}f(x,y,z)dS=$