Hi guys. Thought I'd try and illustrate what I think NonCommAlg is doing above. The integral $\displaystyle \iint_D \left(h-\frac{h}{2a}\sqrt{x^2+y^2}\right)dA$ is that part of the blue cylinder inside of the the red cone: it's the height of the cone h, minus the intersecting surface of the cylinder with the cone times $\displaystyle dA$ over the region $\displaystyle D$ with $\displaystyle D$ being the area of the base of the cylinder. Guess that's confusing though. But once you see that, then it makes sense once you figure what that volume of blue inside the cone is, then just subtract that volume from the total volume of the cone which is $\displaystyle 4/3\pi a^2 h$

Here's the Mathematica code for the first figure in case some are interested (the code for the other two is a bit messy):

Code:

a = 1/2;
h = 1;
cone = RevolutionPlot3D[t/a, {t, 0, a},
BoxRatios -> {1, 1, 1}, PlotStyle -> Red]
cy = Graphics3D[{Blue, Cylinder[
{{0, a/2, 0}, {0, a/2, h}}, a/2]}]
Show[{cone, cy}]