How to tell whether a left and right riemann sum are overestiamtes and underestimates?

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It makes no difference whether the values of a function are positive or negative, if you always choose the smallest value of the function on each interval, the Riemann sum will be an underestimate. If you choose the largest value of the function on each interval, you will get an overestimate:

$$\sum_i \left(\min_{t_{i-1} \le t \le t_i} f(x)\right)\Delta t_i \le \int_a^b f(t)\,dt \le \sum_i \left(\max_{t_{i-1} \le t \le t_i} f(x)\right)\Delta t_i $$

If $f$ is increasing, then its minimum will always occur on the left side of each interval, and its maximum will always occur on the right side of each interval. So for increasing functions, the left Riemann sum is always an underestimate and the right Riemann sum is always an overestimate.

If $f$ is decreasing, this is reversed.

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deezy
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Updated on August 01, 2022

Comments

  • deezy
    deezy over 1 year

    I know that in a positive and increasing function, the right riemann sum is an overestimate and the left is an underestimate, but what about if the function is negative and increasing like this? Which one would be an overestimate and underestimate?

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    • Admin
      Admin almost 6 years
      Did you calculate the corresponding sums? You should be able to see which is bigger.
    • deezy
      deezy almost 6 years
      Wait so the one that is bigger would be an overestimate for this table?
    • deezy
      deezy almost 6 years
      Also, conceptually why would it be an overestimate or underestimate?
    • Admin
      Admin almost 6 years
      To get an idea what happens you could draw a graph and try to understand what the left/right riemann sum actually are.
    • deezy
      deezy almost 6 years
      But the points don't really connect all that well though.
    • deezy
      deezy almost 6 years
      Also, how would you explain it?
    • herb steinberg
      herb steinberg almost 6 years
      Wouldn't it hold for any increasing function (not necessarily positive)? You can always reduce it to a positive increasing function by adding a constant which is less than the left hand value.
  • deezy
    deezy almost 6 years
    So the right sum would be an overestimate because the function is increasing?
  • Paul Sinclair
    Paul Sinclair almost 6 years
    Yes, because $f$ is increasing, it takes on its maximum on the right side of each interval, so it will be greater than or equal to any other Riemann sum on the same partition (or any refinement of that partition).