Symmetry

An odd function is a function $f(x)$ such that $f(-x) = -f(x)$, and an even function one for which $f(-x) = f(x)$. If $f$ is an odd function, then for any $a$,

$$\int_{-a}^a f(x) dx = 0.$$

If $f$ is an even function, then for any $a$,

$$\int_{-a}^a f(x) dx = 2 \int_{0}^a f(x) dx.$$

Both statements are clear if we view integrals as computing the signed area between the graph of $f(x)$ and the $x$-axis.

Example 2.3.7  

$$\int_{-1}^1 x^2 dx = 2 \int_{0}^1 x^2 dx = 2\left[\frac{1}{3} x^3\right]_{0}^1 = \frac{2}{3}.$$