Comparing Right- and Left-Hand Sums
Let f be an increasing function on [a,b] and let R be the region between the graph of f and the x-axis on [a,b].
- Will LHS(n) be an over- or under-estimate of the area of R?
- Will RHS(n) be an over- or under-estimate of the area of R?
(hint: sketch f)
Whatever shape f has, we know f is increasing. This means on any sub-interval f will be smallest at the left endpoint and largest at the right endpoint of that sub-interval:
- Any left-hand sum will be an under-estimate of the area of R. Since f is increasing, a left-hand sum will use the smallest value of f on each sub-interval. The means any left-hand sum will fail to cover all of R.
- Any right-hand sum will be an over-estimate of the area of R. Since f is increasing, a right-hand sum will use the largest value of f on each sub-interval. This means any right-hand sum will cover R and then some.
We see that if f is always increasing then a left-hand sum will give an under-estimate and right-hand sum will give an overestimate. If f is always decreasing then a left-hand sum will give an over-estimate and a right-hand sum will give an under-estimate.
If f alternates between increasing and decreasing, it's possible for both the LHS and RHS to be overestimates, or for both the LHS and RHS to be underestimates.