Upper and lower sums

We introduce two more rectangular approximations.

upper sum approximation

The height of the rectangle is the absolute maximum of f(x) on the subinterval.

lower sum approximation

The height of the rectangle is the absolute minimum of f(x) on the subinterval.

It should be clear that, if the area being approximated has A square units of area, then

lower sum A upper sum

In general, it is rather complicated to compute upper and lower sums. However, if f(x) is monotonic, the situation is much easier. If f(x) is increasing on the interval [a,b], then the upper sum is just the right sum and the lower sum is just the left sum. In the last example with f(x) = x, the right sums (which are upper sums) moved down toward the value of A as the number of subintervals increased. What happens with the left sums (which are lower sums) as n, the number of subintervals, increases? The approximations of the area using these two rules do not generate approximations which are necessarily more or less accurate than the first three rules presented. However, they are informative in that they give lower and upper bounds on what the true area is so that, if the function is not monotonic, at least a range for the true area is available.