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Homework Help: Step functions and integration question

  1. May 10, 2010 #1
    1. The problem statement, all variables and given/known data
    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]

    2. Relevant equations

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

    3. 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?
  2. jcsd
  3. May 10, 2010 #2


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    Science Advisor
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    Gold Member

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