Originally Posted by
chiro Archimedes principle is that (with regards to your problem) the volume of water lost is the volume of the thing that was submerged. In this case its because the water escaped from the cone but in other cases, the water will rise (since it can't escape anywhere).
So you need to calculate the volumes of the water lost relative to the cone and sphere and then use this to get the surface area of the sphere for that particular region.
Another hint is to remember that the region of the integral will be from r = 0 to r = h for some height h (due to symmetry of a sphere) so you will need to match the volume with its corresponding integral and then solve for h. You then use that to get the surface area of the sphere and you're done.