Step functions and integration question

[Trimethyl]

Homework Statement

Consider the function f(x) = x^2 on the interval (0, 1). By considering suitably chosen step functions A and B with partition points at a_j = j/N (0<= j<= N), show that f is integrable on (0, 1) and evaluate its integral. [You may wish to look up a formula for the sum from j=1 to N of j^2]

Homework Equations

the sum from j=1 to N of j^2 is (N)(N+1)(2N+1)/6

The Attempt at a Solution

at first I thought that letting A=(j/N)^2*characteristic function on(a_j-1,a_j]
and B=(j-1/N)^2*characteristic function on(a_j-1,a_j]
might help, but I didn't seem to get anywhere

Can anyone help me find an appropriate step function?

 

[LCKurtz]

Make a partition of mesh size 1/n and look at the difference between the upper and lower rectangular approximations using that partition.