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

Question

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?

Answer 1

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.