1. Definition of an Even Function: A function $f(x)$ is considered even if $f(-x) = f(x)$ for all $x$ in its domain. Geometrically, even functions are symmetric with respect to the $y$-axis.
2. Splitting the Integral: We can split the integral over the symmetric interval $[-a, a]$ into two parts:
$$\int_{-a}^{a} f(x)dx = \int_{-a}^{0} f(x)dx + \int_{0}^{a} f(x)dx$$
3. Property Transformation: For the first part, let $x = -t$, then $dx = -dt$. When $x = -a$, $t = a$; when $x = 0$, $t = 0$:
$$\int_{-a}^{0} f(x)dx = \int_{a}^{0} f(-t)(-dt) = \int_{0}^{a} f(-t)dt$$
Since the function is even, $f(-t) = f(t)$:
$$\int_{0}^{a} f(-t)dt = \int_{0}^{a} f(t)dt$$