7.1. Riemann Integral

Examples 7.1.11(c): Riemann Lemma

Suppose 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?
By assumption 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 -:).

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