(a) If we slice the hemispheres into a number of infinitely thin layers we have to calculate the area of the slice and the horizontal component of the force produced by the pressure.
The radius of the hemisphere = R
At any angle θ:
The radius of the slice r = R Sin θ
Thickness of the slice = R dθ  (measuring θ in Radians)
The surface area of the slice exposed to the atmosphere = dA
dA = (2πr) R dθ
dA = (2π R Sin θ) R dθ
dA = 2π R^{2}Sin θ dθ
The force exerted by the horses is exactly opposed to force exerted by the the air pressure = Air pressure difference X Area
=
Δp X dA
= Δp (2π R^{2} Sin θ dθ)
But only half of this force pulls to the right so it should really be [Δp (π R^{2} Sin θ dθ)], however, the force on the lower right half of the hemisphere is also the same so adding the upper and lower element it goes back to [Δp (2π R^{2} Sin θ dθ)]
The horizontal component of the force for each element ΔF_{h} = [ (Sin θ dθ)] Cosθ
So the total force pulling to the right F_{h} is the Integral of all the elements.
F_{h}


= 

2πR^{2 }Δp 
∫ 
^{π}/_{2} 

Sin θ Cos θ dθ 
0 
Using the identity ∫Sin θ Cos θ dθ = ½ Sin θ^{2}
F_{h}


= 

2πR^{2 }Δp [½ Sin θ^{2}] 
 
^{π}/_{2} 


0 
F_{h}= 2πR^{2 }Δp {(½) [(1) (0)]}
F_{h}= πR^{2 }Δp