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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