Centre of mass of a solid hemisphere (Feynman way)

[saturn_1995]

reference to Feyman lectures vol.1 topic 19.2 locating centre of mass

Feyman gives us the law of pappus to find the centre of mass ,which he proves for semicircular disc and ring.

But when i am trying to extend it to finding the centre of mass of a solid semi-circular solid hemisphere ,i seem to get a different answer from what i get from calculus which is 3R/8.

My approach to problem is using law of pappus extensively:-

1. I assume a solid semi-circular hemisphere into infinite number of semi-circular disc, find each disc's centre of mass using law of pappus.
2. I get a ring with all centre of mass of respective discs.
3. using that ring and law of pappus again i find the centre of mass of that ring.

this value which i find doesn't correspond to 3R/8. is there something wrong in my approach

 

[mathman]

It would help if you showed the numerical details of your calculation.