A tank full of water is in the shape of a cone, with point facing down. The cone is 10 ft deep has 5 feet of water in it. The top of the cone is 3ft in diameter. Write an integral for how much work it takes to raise the water out of the top of the cone. Do not solve. (water is 62.5lb/ft^3)
if the depth is y, the surface of the water has radius 0.15y
so the volume of a slice of water of thickness dy is
v = πr^2 dy = π(0.15(10-y))^2 dy
The work involved in raising 3 ft of water is thus
∫[10,7] π(0.15(10-y))^2 * 62.5 dy
rats - work = weight * distance ...
∫[10,7] πy(0.15(10-y))^2 * 62.5 dy