1. Water The fluid(water) will exert a force on the wall due to
pressure. Pressure in general will be a function of
position over the surface(i.e., a function of x and y).
For the picture shown it will simply be a function of
y in specified coordinate system. For the selected
element, the contribution to the overall force will be
dF = - p dA n = p dA k (n -a unit vector, ┴ to dA in H2O)
If you were to choose another element at a different
location, the pressure might be different and its
contribution to the overall force higher or lower
depending upon its depth. x y
Water dA n
Wall of width W and height H
To find the total force on the wall, we must integrate because pressure is not constant. In this
coordinate system dA = dx dy.
H W
FR =  dF = -   p dA =   p dx dy k
  A  
The pressure must be found in terms of x
and y to do the math: p(y) = Po + gy
From the hydrostatic equation, we know p = p(y), so it is treated as a constant when we integrate w.r.t. x:
H H
FR = W  p dy k = W  [Po + gy] dy k
 
We carry out the integration.
Hence, FR = {PoWH+ gWH2/2} k