The cylinder consists of three surfaces: two flat circular faces (caps) and one curved surface.

1. Curved Surface: The area vector is perpendicular to the surface (radially outward), while the electric field is parallel to the axis. The angle between them is $90^\circ$. Flux $\Phi_{curved} = \int E dA \cos 90^\circ = 0$.
2. Circular Faces: The area vectors for the two faces point outward in opposite directions (parallel and anti-parallel to the field).

• For the face where the field enters, $\theta = 180^\circ$, so $\Phi_{in} = E(\pi R^2) \cos 180^\circ = -\pi R^2 E$.
• For the face where the field leaves, $\theta = 0^\circ$, so $\Phi_{out} = E(\pi R^2) \cos 0^\circ = +\pi R^2 E$.

3. Total Flux: The net flux is the sum of fluxes through all surfaces: $\Phi_{total} = -\pi R^2 E + \pi R^2 E + 0 = 0$.

Alternatively, according to Gauss's Law, the net flux through a closed surface is zero if there is no charge enclosed within it ($q_{enclosed} = 0$). Since the cylinder is merely placed in an external field and encloses no charge, the total flux is zero .