Question

The Deligne Dam on the Cayley River is built so that the wall facing the water is shaped like the region above the curve y=0.4x^2 and below the line y=220. (Here,distances are measured in meters.) The water level is 26 meters below the top of the dam. Find the force (in Newtons) exerted on the dam by water pressure. (Water hasa density of 1000kg/m^3, and the acceleration of gravity is 9.8m/sec^2 .)

Answer 1

To find the force, you multiply density, gravity, area and depth. Because the area and depth are variable, you need to take an integral. Because the damgoes up/down, we'll make the limits of integration y-values, which means our equation must be in terms of y also.
Limits of Integration: The total depth of the water is 194 (220-36). We measure from the bottom up, so limits of integration are 0 to 194.

The depth in the equation is changing, and is equal to (194-y).
You solve the area equation for 'y', so that is sqrt(y/0.4). *You have to multiply the area by 2 so you get both halves of the equation.

Overall, you integrate (9.8)(1000)(194-y)2(sqrt2.5y) from 0 to 194.