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.

Tue Nov 5 14:24:02 EST 1996