The centroid of a hemispherical surface

[Swatch]

I have to find the centroid of a hemispherical surface with radius r.
I want to use Pappus's theorem.S=2*Pi*x*s
The surface S, is 2*Pi*r^2 and the length of the curve s, is Pi*r so the distance from the centroid ,which lies on the y-axis by symmetry, to the x-axis is S/(2*Pi*Pi*r) = r/Pi

This is not the answer by textbook gives for the y coordinate of the centroid. What am I doing wrong? Please help me.

 

[Fermat]

x is the centroid of the generating surface which is a semicircle. And the centroid of a semicircle (a 2D object) is different to that of a hemispherical surface (a 3D object). I'm afraid you can't use Pappus.

 

[Astronuc]

Try this instead - http://mathworld.wolfram.com/Hemisphere.html

I believe the hemispherical is a semi-circle rotated by pi, or a quarter circle rotated by 2pi about an axis corresponding to a radius and corresponding to either end of the arc.

 

[Swatch]

I understand I can't use Pappus's theorem.
So I try to cut the hemispherical shell into horizontal shells up the y axis. Each shell should have the surface 2*Pi*R where R=sqrt(r^2-y^2)
So dS = 2*Pi*R dy
But there is something wrong here, When I integrate from 0 to r I don't get the right answer for the surface of the hemispherical surface.
Could you help me some more, please.

 

[Swatch]

I got it right. I had to calculate the width of the shell.And then integrate.

 

[HURKO]

How can I calculate the centroid of the hemisphere surface using only simple integrals and rectangular coordinates?