# Integration of even (and odd) function

Suppose that $a>0$ and that $f$ is integrable on $[-a,a]$. Show that if $f$ is even then $$\int_{-a}^0 fdx = \int_0^a fdx$$ using the Riemann sum definition of Riemann integrability.

This is what I have tried:

Let $P$ be tagged partition of $[-a,0]$ i.e. $-a = x_0 < x_1 < \ldots < x_k = 0$ and $P^*$ be the symmetrical tagged partition of $[0,a]$ i.e. $0 = x_k < x_{k+1} < \ldots < x_{2k} = a$ where $x_0=-x_{2k}, x_1=-x_{2k-1}, \ldots , x_{k-1}=-x_{k+1}$

Then I try to prove that $S(f;P)=S(f;P^*)$.

Define $S(f;P)= \sum_{i=1}^k f(x_i-1)\cdot(x_i - x_{i-1})$ and $S(f;P^*)= \sum_{i=k}^{2k} f(x_i)\cdot(x_i - x_{i-1})$. Since $f$ is even i.e. $f(x_{2k})=f(-x_0)=f(x_0)$ etc, $S(f;P)=S(f;P^*)$.

Does this proof look okay? Did I use and justify the definition correctly here? Can I do similar proof when f is odd?

Thanks.

-

Details may depend on how the Riemann integral was developed in your course. For one thing, you should quote a result saying that if $f$ is integrable on $[-a,a]$, then it is integrable on subintervals, such as $[-a,0]$ and $[0,a]$.
Then it can go like this: for every $\epsilon>0$ there exists $\delta_1$ such that for every $\delta_1$-fine tagged partition $P$ on $[0,a]$ (that is, a partition with subintervals of length at most $\delta_1$) we have $$\left|\int_0^a f(x)\,dx - S(f,P)\right|<\epsilon \tag1$$ Similarly, there exists $\delta_2$ such that for every $\delta_2$-fine tagged partition $P$ on $[-a,0]$ we have $$\left|\int_{-a}^0 f(x)\,dx - S(f,P)\right|<\epsilon \tag2$$ Now take a $\delta$-fine partition $P$ of $[0,a]$ with $\delta=\min(\delta_1,\delta_2)$. Let $P^*$ be its reflection. Apply (1) to $P$ and (2) to $P^*$. Conclude that $$\left|\int_{-a}^0 f(x)\,dx - \int_0^a f(x)\,dx\right|<2\epsilon \tag3$$ Make the final step based on $\epsilon$ being arbitrarily small.