**Example:** Evaluate
, where

**Solution:** This function looks like this:

We have to decide on partitions. Note that in rectangles that do not involve
the point *x* = 1,*f* is
constant. This means that the supremum and infimum are equal there, in other
words, the upper and lower sums agree at those parts. The only difference
between the upper and lower sum will happen around
*x* = 1,*x* = 1:

We look at the picture

and see that

Thus

The upper and lower sum can be also directly seen from the picture, the shaded area on the left is the upper sum, on the right the lower sum:

We therefore get bounds for infimum and supremum

and so

Consequently,

and

This answer seems clear from the picture, the region consists of three squares of side one.