The ends of a water trough have the shape of the region bounded by the graphs y=x^2 and y=4. To what depth must the trough be fi
lled with water so that the force of the water exerted on either end of the trough is 779.423 lbs? (Water has a density of 62.5 lbs/ft^3. Both x and y are measured in feet.
4 is the maximum depth of the trough so we expect the answer to be less than 4. As the cross section of the trough has the shape y = x², then the width of the water level at any depth y is 2x.
Let a to b be the water depth, σ is the weight density of the fluid (water), y is the depth of water at any level and the width of water at that level is w(x).
The fluid force against the walls of the trough equals :-
b ∫σyw(x) dx. . . . a = 0 and b is the desired depth, y = x², w(x) = 2x a