The resultant force is found by integrating the pressure over the arch surface, S, which means integrating θ from 0 to π and multiplying by the length of the arch.
For the x component:
Integrate or use integral tables to find
R_{x} = 270 ρg
[a/2 sin^{2}θ - b sinθ]_{0}^{π}
Evaluating the integral for R_{x} gives
R_{x} = 0 lb
The y component is
Integrating and evaluating for R_{y}
R_{y} = 270ρg [a/2 (-cosθ sinθ + θ) b cosθ]_{0}^{π}
R_{y} = 270ρg (aπ/2 - 2b) lb
Finally, substitute for the known density, water depth, and arch radius to give
R_{y} = -436,200 lb = -436 kips
Therefore, in vector form, the resultant force is
R = 0**i** - 436**j** + 0**k** kips |