The diagram shows the graph $y = f(x)$, where $f(x)$ is an odd function - HSC - SSCE Mathematics Advanced - Question 5 - 2023 - Paper 1

To find the integral $\\int_{-a}^{a} f(x) \, dx$, we can utilize the property of odd functions.

An odd function $f(x)$ satisfies the condition: $f(-x) = -f(x)$

The integral of an odd function over symmetric limits can be expressed as: $\\int_{-a}^{a} f(x) \, dx = 0$

However, the problem states that the area under the curve from $0$ to $a$ is equal to 1, illustrated by the shaded region (i.e.: $\\int_{0}^{a} f(x) \, dx = 1$). Since the function is odd, we have: $\\int_{-a}^{0} f(x) \, dx = -\\int_{0}^{a} f(x) \, dx = -1$

Thus, the total integral can be computed as: $\\int_{-a}^{a} f(x) \, dx = \left(\\int_{-a}^{0} f(x) \, dx + \\int_{0}^{a} f(x) \, dx \right) = -1 + 1 = 0$

Given this analysis, we can conclude that: The value of $\\int_{-a}^{a} f(x) \, dx$ is 0.