I had recently learned about "Centre of mass" in my physics class. I may be wrong but what I inferred from that lecture was "Centre of mass is more a mathematical concept where we assume a continuous body as a aggregate of infinitely many point masses, and then we assign co-ordinates to each of the points then, the Centre of mass's co-ordinate is the average of all the X and Y coordinates of all the points.
Say, I have three points in the 2d plane (0,0), (1,1), (2,2).
C.O.M = (1,1)
X co-ordinate of com = sum of all x co-ordinates / no. of co-ordinates
vice versa for Y co-ordinate.
Now, problem arises when I tried to find COM of a continuous system.
example :-
f(x) = x : x ranges from 0 to 1
I know the COM of this system should be (0.5,0.5) using my knowledge from the physics lecture, but when I tried to use the definition I had arrived i.e., "the Centre of mass's co-ordinate is the average of all the X and Y coordinates of all the points." I realized that I would have to take the average of the Infinite no. of points that lay on that line.
X co-ordinate of com = sum of all x co-ordinates / (no. of co-ordinates) <--- does this
become infinity ( I highly doubt it) or maybe it converges to something.
How do I do this?
Use formula: $$\text{Mean}={1\over{b-a}}\int_a^b{f(x)dx}$$ Ex. $y=x^2$ $$M={1\over{b-a}}\int_a^b{x^2dx}=(a^2+ab+b^2)/3.$$