ok...i did this way.. begin by drawing the FBD of the mass m which is surrounding the inner cylinder... considering a thin slice of the bucket as i drew in the diagram...let the mass of each triangular slice be dm... then on each slice a force N will act...and the vertical component of this will balance mg and the horizontal component will balance the force F due to the cylinder...hence the net downward force due to this dm is its own weight...and by radial symmetry the net force on the curved surface is the weight of this light blue portion...the volume can easily be found using geometry and then multiply by dg to get the weight...
PS: do rate me for the diagram if u wish...it took some work..anyways i was bored...