Let's say we have a water trough and the ends of the water trough have the shape of the region bounded by the graphs $y=x^2$ and $y=4$, with both $x$ and $y$ measured in feet. How deep would I have to fill the trough with water so that the force exerted by the water on either end of the trough is 779.423 lb? Water has a density of 62.5 pounds per cubic feet.
I know that I have to use the shape of the ends of the water trough to find the area of the end of the trough, so if I filled the trough to let's say 2 feet deep, I would solve for $\int_{0}^{2} (\sqrt x)dx$.
How would I solve this problem? Also I know that the answer is between 2 feet and 4 feet.