To find the average y-value of a function, f(x), on interval (a,b), simply take the integral of f(x) from a to b, and divide by (b-a). To find the average x-value of a function (basically the continuous counterpart of the discrete mean of a set of data), you must take the integral of x*f(x) from a to b, and divide that quantity by the integral of f(x) from a to b.
That stuff is self-explanatory, and easy to see with some basic calculations. What I do not understand is why the average x-value of a function is not equal to the average y-value of its inverse.
Take for instance the function f(x)=x2. Its average x-value on the interval (0,4) is Int(x3) from 0 to 4, divided by Int(x2) from 0 to 4. This value is 3; thus, the mean of x2 from 0 to 4 is 3. Its inverse is f-1(x)=sqrt(x). To find its average y-value, you do Int(sqrt(x)) from 0 to 16 (since f(4)=16) divided by 16. This value is 2.66666... I do not understand why this value is not equal to the average x-value of the original function.
To throw some more info out there: I know that the mean of a function is where it balances about that point, i.e. it is a 2-dimensional "mass" with constant density, and it balances at its average x. So, f(x)=x2 on (0,4) balances at x=3. But if you are to find the average distance of the function to the y-axis on that interval, it is 2.66666...
Any clarification?
That stuff is self-explanatory, and easy to see with some basic calculations. What I do not understand is why the average x-value of a function is not equal to the average y-value of its inverse.
Take for instance the function f(x)=x2. Its average x-value on the interval (0,4) is Int(x3) from 0 to 4, divided by Int(x2) from 0 to 4. This value is 3; thus, the mean of x2 from 0 to 4 is 3. Its inverse is f-1(x)=sqrt(x). To find its average y-value, you do Int(sqrt(x)) from 0 to 16 (since f(4)=16) divided by 16. This value is 2.66666... I do not understand why this value is not equal to the average x-value of the original function.
To throw some more info out there: I know that the mean of a function is where it balances about that point, i.e. it is a 2-dimensional "mass" with constant density, and it balances at its average x. So, f(x)=x2 on (0,4) balances at x=3. But if you are to find the average distance of the function to the y-axis on that interval, it is 2.66666...
Any clarification?