Is the right Riemann sum an overestimate or an underestimate?
Understand the Problem
The question is asking about the properties of the right Riemann sum in relation to its accuracy in estimating the area under a curve. It is generally understood that whether it underestimates or overestimates depends on the behavior of the function being integrated. For increasing functions, it will overestimate the area, while for decreasing functions, it will underestimate it.
Answer
A right Riemann sum is an overestimate if the function is increasing and an underestimate if the function is decreasing.
The final answer is that a right Riemann sum is an overestimate if the function is increasing and an underestimate if the function is decreasing.
Answer for screen readers
The final answer is that a right Riemann sum is an overestimate if the function is increasing and an underestimate if the function is decreasing.
More Information
For an increasing function, the right rectangles will always overshoot the actual area under the curve, leading to an overestimate. Conversely, for a decreasing function, the right rectangles will undershoot the actual area, resulting in an underestimate.
Tips
Avoid the common mistake of not considering the behavior of the function (whether it is increasing or decreasing) when determining if a Riemann sum is an overestimate or underestimate.
Sources
- How to tell whether a left and right riemann sum are overestimates and underestimates - math.stackexchange.com
- Determining Whether an Approximation of a Definite Integral is an Overestimate or Underestimate - study.com
- Riemann Sums | Teaching Calculus - teachingcalculus.com
