Ione

I have been thinking about this for the past while. Basically, what you did is you put the the base in the z=0 plane, and then uintegrated over x, using

with the volume of each slice being 2y by 2y by dx. Which as I see it is correct, provided that each slice is the same height. Each slice has to have the same height by symmetrtry, as It doesn't matter when you rotate the object. This gives the total volume as

.

However, the volume of a cylinder is

. Taking our h to be twice the radius (2r) then gives the volume as 2*pi*a^3, which means that it has two different volumes, or something that I am thinking of is wrong. Is the object described not a cylinder? Which actually doesn't change the problem, because when you use your method to find the cylinder volume, you get 23*y*hdx, which still does not give any pi dependance. So what am I missing?