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Proving a Function Is Riemann Integrable

  1. Mar 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Here is a link to the problem which I am currently working on:

    http://math.umn.edu/~jara0025/Math4603/Math4603H9Answers.pdf [Broken]

    2. Relevant equations

    3. The attempt at a solution

    The specific problem I am working on is found on page 3, and is the first problem on that page. The step I am having trouble with is

    $$= \sum_{i=0}^{n-1} (x_{i+1} - x_i)(x^2_{i+1} + x_{i+1}x_i + x_i^2) (x_{i+1} - x_i)$$

    $$ < \delta 3 \sum_{i=0}^{n-1} (x_{i+1} - x_i)$$

    What is going on?
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Mar 25, 2015 #2


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    The point is that ##x_i \le 1## and ##(x_{i+1}-x_i) \lt \delta## for all ##i##. Try using those inequalities.
    Last edited by a moderator: May 7, 2017
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