Calculating Work Required to Pump Water from a Hemispherical Tank

[mystic3]

Homework Statement

A tank in the form of a hemispherical bowl of radius 4 m is full of water. Find the work required to pump all of the water to a point 2 m above the tank.

Homework Equations

w = fd = mgd
density = 1000L/m²
1L = 1kg

The Attempt at a Solution

x = height

w = V*d*g*h
= (π)(r^2)(x)*(1000)*(9.8)*(x)*dx

I think my problem is trying to show how the height a strip of cylinder varies with the radius. I've tried using similar triangles but it was obviously incorrect. Is there a similar property I can use for hemispheres?

 

[Mark44]

Think about this in terms of horizontal layers of water. It's probably most convenient to put the origin at the top level of the hemispherical tank, so that \Delta y ranges from -4 to 0. Each layer has to be lifted (pumped) from its position in the tank to a point 2 m. above the top of the tank. You are using x in your integral expression; I would use y, and I would also find the equation of the semicircular cross section of the tank so that I could exploit a relationship between x and y at a point (x, y) on the semicircular boundary of the tank. A sketch will be very helpful if you haven't already done one. A drawing of a half-circle will do just fine, as long as you realize that the water layers are three-dimensional.

 

[mystic3]

w = V*d*g*h <--new

V = pi*r^2*dh
r^2 = 8h-h^2
d = 1000
g = 9.8
h = 6-h

V = 9800pi ∫ [(8h-h^2)*(6-h)] dh
V = 9800pi*(24h^2 - (14/3)h^3 + (1/4)h^4)
V ~ 1463466pi Joules

This is the first time I've done a question like this and the energy seems a little high to me, is it correct?

I know I can't check it with the potential energy equation because you need different amounts of energy to lift the water at different heights, is there anyway I can check that this is correct?