Here's a diagram of the drum half full lying horizontally.
A point on the circle an angle a from the horizontal has the coordinate (2cos(a), 2sin(a)). The horizontal distance from the origin is 2cos(a) so the width of the circle is 4cos(a).
Note that in your function S(x) the variable x is the vertical distance.
You have the function for distance as a function of angle, now find it as a function of x
Also remember when doing the integration that the 0 point for x is the centre of the circle not the ground point.