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 $\displaystyle x_{cm}=\frac{1}{M}\int{x}\ dm$ and

$\displaystyle y_{cm}=\frac{1}{M}\int{y}\ dm$

where $\displaystyle x_{cm}$ and $\displaystyle y_{cm}$ refer to the centre of mass along the respective axis, $\displaystyle M$ is the mass of the plate, and $\displaystyle dm$ 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

$\displaystyle y_c=\frac{\int_{-A}^{A}(A^2-x^2)dx}{\pi \cdot A^2}-\frac{\int_{-a}^{a}(a^2-x^2)dx}{\pi \cdot a^2}=\frac{\frac43 a^3}{\pi \cdot a^2}=\frac{4}{3\pi}(A-a)$