# what does it mean to take average of something that is continuous?

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?

• Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Commented Jun 29, 2023 at 7:11
• The centre of mass has a physical meaning and does not need a object to be made up of atoms to find it. At a mathematical level it may require integration (in a sense the limit of summation). Commented Jun 29, 2023 at 7:37
• the beauty of calculus Commented Jun 29, 2023 at 19:41

Use formula: $$\text{Mean}={1\over{b-a}}\int_a^b{f(x)dx}$$ Ex. $$y=x^2$$ $$M={1\over{b-a}}\int_a^b{x^2dx}=(a^2+ab+b^2)/3.$$
• The result is $(a^2+ab+b^2)/3$ instead of $b^3/2-a^3/2$. Commented Jun 29, 2023 at 19:18