Electric Field of a charged sphere with cylindrical gaussian surface

So the problem statement is:



A conducting solid sphere (R = 0.167 m, q = 6.63·10–6 C) is shown in the figure. Using Gauss’s Law and two different Gaussian surfaces, determine the electric field (magnitude and direction) at point A, which is 0.00000100 m outside the conducting sphere. (Hint: One Gaussian surface is a sphere, and the other is a small right cylinder.)



Now with the spherical gaussian surface i've got no doubt..



The problem with the cylindrical surface is that the electric field it's not constant ,is it?

in some parts the "area vector" is parallel to the vector of electric field and in some part there's an angle.



So i'm not sure if i have to find a function of the electric field with respect to the radius of the sphere or how to get ride of this problem.



What i've first tried was to "assume the electric field" was constant so integrating dA what i've got:

since the height of cylinder is h=2r (of the sphere).

and there's flux all over the cyinder: E(2pi*h+2pi*r^2)=Q/ε...

E(4pi*r+2pi*r^2)=Q/ε.... and so E=Q/(4pi*r+2pi*r^2)ε... but i'm not sure if a i could assume that...