You essentially have the problem of finding the induced surface charge density on a long, cylindrical conductor placed in a uniform E-field oriented perpendicular to the axis of the cylinder. This is a fairly standard boundary value problem in electrostatics using separation-of-variables in Laplace's equation. The solution will give you something similar to what is given in equation (3)

owever, the presence of ##i## in this equation looks odd to me. But, thankfully, this ##i## disappears in equation (5).
##\sigma(\theta)## is supposed to be the "charge per unit length". I'm not sure how to interpret this. If you include an additional factor of the length, ##L##, of the cylinder, then the equation (without the ##i##) would be
##\sigma(\theta) = 2rL\epsilon_0\left(E_x \sin\theta + E_y \cos \theta \right)##
I believe this now represenst the charge per unit angle ##\theta##. Then it would make sense to write ##q_a = \int \sigma(\theta) d \theta## as in equation (4).