What is the centroid of a hollow spherical cap? I have a unit hollow sphere which I cut along a diameter to generate two equivalent hollow hemispheres.  I place one of these hemispheres on an (x,y) plane, letting it rest on the circular planar face where the cut occurred. 
If the hemisphere was solid, we could write that its centroid in the above case would be given as $(0,0,\frac{3}{8})$.  Given that the hemisphere is hollow, can we now write its centroid as  $(0,0,\frac{1}{2})$?

Oddly enough, there is a much faster way to verify that the centroid
of a hollow spherical cap arranged, as the poster said, is at $(0,0,1/2)$.
If the cap is infinitesimally thin-walled, then we need only look at
its area $A$ and assume it has a mass $M$ evenly spread over that
area. The area $A$ (~$M$) of the cap is $A = 2 \pi R h$, where $h$ is the
height of the top of the cap above its base, in this case its base
is the base of a hemisphere, so h equals $R$. But this thin shell
can be divided by a horizontal plane into two regions, each having
an $h$, or a thickness, of $R/2$: one region being a smaller cap
resting on the upper base of a "spherical segment," whose lower
base is the original base of the hollow hemisphere.
The area of the smaller cap is now $A_1 = 2 \pi R (R/2)$.
By definition (see Wolfram, wikipedia..) the area of the lower
"segment," excluding the area of its bases, is $A_2 = 2 \pi R (R/2)$.
These are clearly identical areas which also correspond to
identical masses. I.e., the geometric centroid, which is the
same as the centroid of mass, is the point on the z-axis
lying on the horizontal dividing plane, which is $R/2$ above
the origin.
