Calculus/Physics Question

I was recently given a problem involving work done with a variable force:

A hemisphere-shaped tank that has a radius of 10 is filled with Benzene which has a fluid density of 100. How much work is done pumping the liquid out of the top.

I set up my integral, but I am lacking a factor that represents the area of each “layer”. The problem was deriving a solution with respect to the depth. I assumed that each layer would have a radius equal to the depth, thus the area of each layer is pi * radius^2. However a different answer was given. Am I wrong in this assumption and how can I prove it?

Well, to figure out the area of each “layer,” you need to know the radius of the layer. I will assume that the hemisphere positioned like a bird-bath, with the largest layer on top. In that case, the distance of the layer from the top, squared, plus the radius of the layer, squared, equals the square of the radius of the hemisphere. Think about it - those three items form a right triangle.

You can solve the above equation for the radius of the layer, then square that number and multiply by pi and you should get a reasonably integrable formula.

If you want more insight into this issue, you might look up the derivation of the formula for the volume of a sphere.