Find Center of Mass of Homogeneous Semicircular Plate

[team31]

Find the center of mass of a homogeneous semicircular plate, let R be the radius of the circle.
I have think I need to use ingergration to slove this problem, I'm stuck with that R is also changing as if I'm cutting little slices, so does that mean i have to set up two intergra, one for R solve that, and put it into another intergra which is for Ycm?

another question is there is a cube box with no top lays on a coordinate system x,y,z, z is vertical. and the side is 40 cm, I can find xcm=20 and ycm=20 very easily, but i had a hard time to find z, and i used harder way to find z, which is tp set zcm=x, and calculate the weight between the top half of the box and bottom half of the box,
40(40-x)+2(40-x)2=20*20+40x+40x.
from there I got Zcm=17.5, and that is not even the right answer, can u guys suggest a better and easier way for me to slove it?

 

[phucnv87]

$$X=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{xdm}{M}$$

 

[team31]

huh, i got it, 4r/3pi is my answer, is that right?

 

[team31]

could anybody give me a hint on my question #2, please