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(D) The cylinder has three surfaces: two circular end faces ($A$ and $B$) and one curved surface $(C)$.For the circular face $A$,the area vector point

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(D) The cylinder has three surfaces: two circular end faces ($A$ and $B$) and one curved surface $(C)$.For the circular face $A$,the area vector points outwards (opposite to the electric field),so the flux is $\phi_A = \vec{E} \cdot \vec{A} = E(\pi R^2) \cos(180^\circ) = -E\pi R^2$.For the circular face $B$,the area vector points outwards (in the direction of the electric field),so the flux is $\phi_B = \vec{E} \cdot \vec{A} = E(\pi R^2) \cos(0^\circ) = +E\pi R^2$.For the curved surface $C$,the area vector at every point is perpendicular to the electric field,so the flux is $\phi_C = \int \vec{E} \cdot d\vec{s} = \int E ds \cos(90^\circ) = 0$.The total flux through the cylinder is $\phi_{total} = \phi_A + \phi_B + \phi_C = -E\pi R^2 + E\pi R^2 + 0 = 0$.

$A$ cylinder of radius $R$ and length $L$ is placed in a uniform electric field $E$ parallel to the cylinder axis. The total flux for the surface of the cylinder is given by

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