Examples from Double integrals and Riemann sums

A midpoint approximation

Estimate the signed volume under

over

by dividing $R$ into $4\times2$ rectangles and using their midpoints.

Because $f$ is negative on part of this rectangle, the result estimates a signed volume, not the geometric volume between the graph and the $xy$-plane. For instance, $f(-\tfrac32,1)=-\tfrac12$, so that box contributes negatively.

The subinterval widths are $\Delta x=1$ and $\Delta y=2$, so $\Delta A=2$. The midpoint $x$-values are

and the midpoint $y$-values are $1$ and $3$. For $y=1$, the four function values sum to $4$; for $y=3$, they sum to $36$. Therefore

Upper and lower sums on eight squares

Partition

into eight squares of side length $\tfrac12$ and let

The function is increasing in both variables on $R$, so on each small square the minimum is at the lower-left corner and the maximum at the upper-right corner. Every square has area $\tfrac14$, so

Thus

Refining the partition narrows the gap.

A function that is not integrable

Define

Every rectangle contains points where both coordinates are rational and points where at least one is irrational. Hence on every small rectangle,

For every partition $P$,

The upper and lower sums never approach one another, so $f$ is not Riemann integrable.