Help finding average of the square of number range.

[Delpen9]

I want to find the average of the square (^2) of every fraction between 1 and 0. Someone told me that is was 1/3rd, and that makes sense because I'm using this to derive the volume of a cone (1/3*pi*R^2*H).

May someone explain how to do this without notations or calculus related symbols? Moreover, I just need the answer extremely broken down.

 

[Ishuda]

I want to find the average of the square (^2) of every fraction between 1 and 0. Someone told me that is was 1/3rd, and that makes sense because I'm using this to derive the volume of a cone (1/3*pi*R^2*H).

May someone explain how to do this without notations or calculus related symbols? Moreover, I just need the answer extremely broken down.

I assume you mean rational fraction so I will assume that (also). Given a ration fraction A greater than 0 and less than 0.5. Let x be any number satisfying
0 < x < A < 0.5
and let
y = 2 A - x
Since A and x are rational fractions, so is y. Also, since A and x are positive, y is less that 2A and
2 A - y = x < A
or y is greater than A. Thus, for every rational fraction x between 0 and A, there is a corresponding rational number greater that A and less than 2A and their average is
$\displaystyle \frac{x+y}{2} = \frac{x + 2 A -x}{2} = \frac{2 A}{2} = A$

Thus the average of all rational fractions between 0 and 2A is A for any A between 0 and 0.5. It is left to the student to prove this for the square of the rational fractions.

Now the mean is another matter altogether

 

[Ishuda]

I want to find the average of the square (^2) of every fraction between 1 and 0. Someone told me that is was 1/3rd, and that makes sense because I'm using this to derive the volume of a cone (1/3*pi*R^2*H).

May someone explain how to do this without notations or calculus related symbols? Moreover, I just need the answer extremely broken down.

Just what can one use? For example, could one use the sum of the first n squared integers divided by n cubed approached 1/3 as n became large?