Find the center of mass of the thin semicircular region of consant density,sigma=4 bounded by the x-axis and the curve y=(121-x^2)^1/2
The center of mass is simply finding the the mass of the object = density * volume and finding the displacement of the mass to the x-axis and y-axis to find the center of mass, which is usually pair (x',y')
However, This link can explain better than I can
Pauls Online Notes : Calculus II - Center of Mass
The computation of the coordinates of the 'center of mass' of a thin region with density is perfomed is succesive steps...
a) first is computed the 'whole mass' as...
(1)
b) then the coordinates of the 'center of mass' are computed as...
(2)
In our case is constant so that it can be and is an half circle of radious and also for it we can suppose . Now we can compute first ...
(3)
... and then...
(4)
(5)
Kind regards