Question

Is the right Riemann sum always an overestimate? Why?

a. Yes, because it always includes the maximum function values
b. No, it can be an underestimate in certain cases
c. Yes, due to the nature of the Riemann sum method
d. No, it depends on the function and the interval

Answer 1

The right Riemann sum is not always an overestimate; it can overestimate or underestimate the value of an integral depending on whether the function is increasing or decreasing on the selected interval.

Step-by-step explanation:

The right Riemann sum is not always an overestimate for the integral of a function over a given interval. The right Riemann sum approximation depends on the behavior of the function over the selected interval. If the function is increasing on the interval, then the right Riemann sum will overestimate the actual integral because it will select the maximum values of the subintervals. Conversely, if the function is decreasing, the right Riemann sum will underestimate the actual integral since it will select values that are less than the actual function values over the interval. Therefore, the correct answer is d. No, it depends on the function and the interval.