Work/Calculus problem

Question

A certain open top container is made by rotating the curve y = x^2/a about the y-axis from x = 0 meters to x = 3 meters, where a = 1 meter. Initially the container is filled to a height of H = 4 meters with water of density ρ = 1000 kg/m^3 . How much work does it take to pump all of the water out of the container? Take g = 10 m/s^2

Answer 1

Work = integral [ (9 -y) dm g ]  : limits  0 to 4

Where the infinitesimal of mass  dm is given by dm = pi x^2   rho dy

But  x^2 = a y so

Work = integral [ (9-y) g rho pi a y dy ] : limits  0 to 4

The integral of y is easy to carry out resulting in

Work = pi a g rho [ (9/2) y^2 – (1/3) y^3 ] evaluated at upper limit of y =4

Work = pi a g rho 152/3

Plugging for pi, a, g and rho gives

Work = 1591 kJ