I'm given the following problem:

Prove that if $f$ is negative and increasing between $a$ and $b$, then the left Riemann sum is an underestimate and the right Riemann sum is an overestimate.

After drawing a graph, I concluded that this statement is false but I do not know how to prove this algebraically. Any pointers?

  • $\begingroup$ It is actually true; you must be confusing something! $\endgroup$ – RGS Jan 17 '17 at 0:12
  • $\begingroup$ Remember that for functions that are negative, area and integral aren't quite the same thing. $\endgroup$ – Arthur Jan 17 '17 at 0:14
  • $\begingroup$ image.prntscr.com/image/277120c505564af2bd5fb0301d03d325.png $\endgroup$ – Question asker Jan 17 '17 at 0:16
  • $\begingroup$ It seems to me that there is an overestimate for the area of a left reimann sum in this graph, or is is that the area of a negative function is negative, meaning that it is less area in the end? $\endgroup$ – Question asker Jan 17 '17 at 0:18
  • $\begingroup$ @Questionasker The left sum overestimates the area, but underestimates the integral. $\endgroup$ – Arthur Jan 17 '17 at 0:19


Consider the partition $[x_0, x_1, x_2, \cdots, x_n]$ of $[a, b]$ where $x_0 = a, x_n = b$;

Notice that $f(x_0) \leq f(x_1) \leq \cdots \leq f(x_{n-1}) \leq f(x_n)$. (the $\leq$ may be exchanged by $<$ if the problem statement means that $f$ is strictly increasing.

Are you able to write the expressions for the left and right riemman sums of that partition?


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