Calculating center of mass integral

Question

I have a few problems with the calculation of the center of mass of this curve,

Can someone explain me the first two steps?

Thanks

Answer 1

The length of this curve is the quarter of circle perimeter $$=\frac {2\pi R}{4}=\frac {\pi R}{2} $$

mass=density x length. $$M=\rho .\frac {\pi R}{2} $$

the abscisse of a point $M $ in the curve is

$$x=R\cos (\theta )$$

where $\theta $ is the angle $(\vec{Ox},\vec {OM}) $.

For $y_G $, you will have $$\frac {1}{M}\int_0^R \rho R\sin (\theta)Rd\theta $$