Final answer:
The right Riemann sum can be an overestimate when dealing with decreasing functions because the highest value at the right endpoint will make the estimated area larger than the actual area. For increasing functions, it is more likely to underestimate. Accuracy of the estimate improves with tighter bounds and fewer subintervals.
Step-by-step explanation:
Whether the right Riemann sum is always an overestimate depends on the nature of the function being approximated. When estimating the area under a curve on a particular interval using a Riemann sum, the type of function dictates whether the approximation is an overestimate or an underestimation. For a function that is decreasing on the interval, the right Riemann sum will typically overestimate the actual area, because the value at the right endpoint of each subinterval will be higher than the actual function value on the majority of the subinterval. Consequently, the rectangles formed by the right Riemann sum will sit above the curve, leading to the overestimation.
Conversely, if the function is increasing over the interval, the right Riemann sum tends to be an underestimate, as the function's value grows higher than the right endpoint's value for the remainder of the subinterval. In practice, obtaining an accurate estimate involves tightening the bounds of the subintervals. The fewer and narrower the subintervals, the closer the Riemann sum will be to the true area under the curve. When calculating estimates, maintaining simplicity by using one significant figure can often be sufficient for a rough estimate. Ensuring that results are reasonable and consistent with known values is crucial for validating the accuracy of an estimate.
It is important to remember that these generalizations about overestimation assume that the function is either increasing or decreasing; for functions that are not monotonic on the interval, more careful analysis is needed to determine whether the right Riemann sum will be an overestimate or not.