## Examples 7.1.11(c): Riemann Lemma

Suppose

By assumption *f*is Riemann integrable over an interval*[-a, a]*and*f*is an odd function, i.e.*f(-x) = -f(x)*. Show that the integral of*f*over*[-a, a]*is zero. What can you say if*f*is an even function?*f*is Riemann integrable so we can use the previous example and take a sequence of partitions with smaller and smaller mesh, compute a Riemann sum for each partition, and find the limit.

For an evenly spaced partition that includes 0 it is easy to compute a particular Riemann sum (such as a "middle Riemann sum") as long as the function is odd.

The rest is left as an exercise.

For even functions, i.e. functions where *f(x) = f(-x)*, we can
show that the integral of *f* from *-a* to *a*
is twice the integral from *0* to *a*. The details are
left as an exercise again (sorry -:).