Calculating work in a hemispherical tank

[Mangoes]

This actually is from a calculus course, but the problem's a physics application. I was unsure if whether or not I should ask this in the math section or the physics section, so apologies if this doesn't exactly fit in here.

Homework Statement

"A hemispherical tank of radius 6 feet is positioned so that its base is circular. How much work is required to fill the tank with water through a hole in the base if the water source is at the base?"

The answer is wanted in foot-pounds and water is given to weigh 62.4 pounds per cubic foot.

The Attempt at a Solution

Work is the product of distance and a force.

To express force:

If I made a slice of a height of Δy, it would have a volume of pi*r^2*Δy. The radius is also equivalent to the x-coordinate of the semicircle. However, since I'll eventually be integrating with respect to y, I'd like to write the radius in terms of y.

Choosing to place the origin at the center of my hemisphere, http://i.imgur.com/eacNB.png

$$x^2 + y^2 = 36$$
$$x^2 = 36 - y^2$$
$$x = - \sqrt{36 - y^2}$$

Plugging this in as the value of my radius gives the dimensions of slice in terms of y. I'd like to approximate the force in this volume, so I multiply the dimensions by 62.4 (the force of water per cubic foot)and I'll use that expression to express force.

$$ΔF = (62.4\pi)(36 - y^2) Δy$$

Now, the distance, d, required for the water to move from the base to some level of the tank is d = y + 6. My reasoning here is that at the lowest level of the tank, y = -6, the distance would be 0 and it'll increase as y goes towards 0.

$$Δw = (62.4\pi)(36 - y^2)(y + 6) Δy$$

As Δy approaches 0,

$$w = \int (62.4\pi)(36 - y^2)(y + 6) dy$$

I don't know how to write limits of integration on here, but I'd say I'd have to integrate from -6 to 0.

I didn't feel like actually going through the algebra, but when I evaluate the definite integral through wolframalpha, the number that comes out doesn't match the answer.

As far as I can tell, there isn't any error with units and I can't see what I'm doing wrong. I've done a couple of these problems already, but this is my first one with a hemispherical tank, so the shape might be the source of my problems.

What am I doing wrong?

 

[jbsteele]

The main issue with your attempt at a solution is that you have not considered the pressure of the water when calculating the work. The pressure of the water is an important factor in determining the amount of work required to fill the tank. When you integrate, you should be integrating the pressure of the water (in this case, 62.4 pounds per cubic foot) multiplied by the distance moved (which, in this case, would be y + 6). This will give you the total work required to fill the tank.