Bonus: Find the ratio of volumes if the solids are cut into 3 equal lengths along h.

First, let us look at the pyramid. If we slide the axis of its height on the x-axis, we have a sloping line with cross-sections as squares. The height is on the x-axis and the largest square at x = h, is a/2. The slope of the line is a/2h using rise over run. Thus we have to integrate the area of the cross section from 0 to h. This area is (2y)² or (ax/2h) ². If we integrate this we get a²h/3. So as we know, the volume of a square pyramid is a²h/3. Now integration from 0 to h/2 and from h/2 to h gives a 1:7 ratio for the volumes.
The same procedure is applied for a cylinder. The volume becomes pi*r²h/3 and the ratio is indeed 1:7 again.
For the bonus, we simply do the integral from 0 to h/3, h/3 and 2h/3 and 2h/3 to h. Comparing these values, for both cylinders and pyramids, we obtain a 1:7:19 ratio.