Centre of Mass by integration

I've come across this question (which looks more like physics than maths, to be honest), and I really need some help...

Q. Find the centre of mass of a homogenous semi-circular plate. Given that and

where and refer to the centre of mass along the respective axis, is the mass of the plate, and is the small change in mass.

Oh, by the way, if there's a type of solution apart from calculus (geometric or otherwise), I'd be glad to know of it...

Centre of mass by integration semi-circle with thickness

Supposing that you wanted to find the centre of mass of a semi-circle plate with radius A and this semi circle had a smaller semi-circle plate with radius a removed then would the centre of mass be equal to