Let *f* be a decreasing 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*?

Answer

Whatever shape *f* has, we know *f* is decreasing. This means on any sub-interval *f* will be largest at the left endpoint and smallest at the right endpoint of that sub-interval:

Any left-hand sum will be an over-estimate of the area of *R*. Since *f* is decreasing, a left-hand sum will use the largest value of *f* on each sub-interval. The means any left-hand sum will cover all of *R* and then some.

Any right-hand sum will be an under-estimate of the area of *R*. Since *f* is decreasing, a right-hand sum will use the smallest value of *f* on each sub-interval. This means any right-hand sum will fail to cover *R*.