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# Definite Integrals

# Comparing Right- and Left-Hand Sums

### Sample Problem

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*)

Answer.

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.