Fluid Statics: Find vertical and horizontal forces in tank

[zealeth]

Homework Statement

The tank shown in the figure is filled with water and fastened to the ground. There is a small opening on top of the semi-spherical part of the tank.
a) What is the magnitude of the horizontal force on the screws? Explain using a free body diagram.
b) Consider the proper area element (dA) and determine the vertical force in the screws. Assume the weight of the container can be neglected.

Homework Equations

p = ρ*g*h
F_H = p_c dA = ∫_Ap dA
F_V = ρg∀ or F_V = dF_y
(∀ = Volume)

The Attempt at a Solution

Note that you are only given what is seen in the image, and the tank is completely filled.

a) F_H = p_c * A = ∫p dA, p = ρgr, A = 1/2 π r², dA = 2πr dr
∫₀^R 2ρgr² dr = (ρgπR²)/4
I then incorrectly defined h in my free body diagram and used that to calculate F_H for the lower part of the tank, which is apparently not needed.
The solution provided for this problem is that all F_H will cancel out, so F_H = 0 (this is the entirety of the solution provided, no other information was given). Why is this?

b) F_V = ρg∀, ∀=(4/3)πR³ * (1/2) = 2/3πR³, F_V = (2/3) ρgπR³. I came up with this solution after once again incorrectly defining h for the height of the cylinder part of the tank, but why is h not needed here? Why is there no vertical force in the screws from the cylinder?
The solution provided is seen below, with the correct answer being F_V = (1/3) ρgπR³ (warning: large image). Why is the integration from 0 to π/2, and not 2*∫₀^π/2?

Spoiler

Any help is very much appreciated.

 

[proton007007]

the cylinder is hollow and filled with water which applies a perpendicular horizontal force on it only . there is no vertical force due to the cylindrical water . net Horizontal force due to cylindrical part is zero as every small force dF applied on one side (say towards the right of the figure) is canceled out by the diametrically opposite side's dF of the same magnitude(that is the force towards the left ) .
hence force on screws will be determined by the spherical part only as it is given in the solution .

proton007007