Find the center of mass of a semi-circular plate

[Debelius]

Homework Statement

Find the center of mass of a semi-circular plate of radius r

Find the volume when the plate (above) is rotated around a line along its straight side

Homework Equations

2(pi) integral of r dr

The Attempt at a Solution

I honestly don't know how to do centroids. :-( I'd like to know how to actually solve this problem.

 

[Knissp]

The x coordinate of the center of mass should be zero (intuitively).
For the y coordinate, assuming that the mass is uniformly distributed, use this equation (which can generalize for x):
ycentroid = \frac{\int\int_R y dA}{A}

The volume is thus computed from the theorem of Pappus which states that
volume = (area of R) * (distance traveled by the centroid)
where the distance traveled by the centroid is 2*pi*ycentroid

Hope that helps, let me know if you need further clarification.
(Also, the volume when rotated about the straight side should come out to be that of a sphere, which is 4/3*pi*r^3, so you can use that to make sure you did it right.)