Consider the functions $y = f(x)$ and $y = g(x)$, and the regions shaded in the diagram below - HSC - SSCE Mathematics Extension 1 - Question 2 - 2024 - Paper 1

To find the total area of the shaded regions, we must integrate the difference between the two functions, $f(x)$ and $g(x)$, across the intervals where the regions are defined in the diagram. Given the regions shaded above, we need to consider the parts where $f(x)$ is above $g(x)$ and vice versa. The area can be comprehensively calculated by breaking down the individual intervals:

1. From $-4$ to $-3$, the area is given by the integral of $g(x) - f(x)$ since $g(x)$ is above $f(x)$.
2. From $-3$ to $-1$, the area is $f(x) - g(x)$ since $f(x)$ is above $g(x)$.
3. From $-1$ to $3$, $f(x)$ remains above $g(x)$, thus we again integrate $f(x) - g(x)$.
4. Finally, from $3$ to $4$, the area is $g(x) - f(x)$ as $g(x)$ is above $f(x)$.

Therefore, the combined expression for the total area of the shaded regions is represented accurately by option D:

$\int_{-4}^{-3} g(x) - f(x) \, dx + \int_{-3}^{-1} f(x) - g(x) \, dx + \int_{-1}^{3} f(x) - g(x) \, dx + \int_{3}^{4} g(x) - f(x) \, dx$

This representation captures all the aspects of the shaded regions, validating that option D is indeed the correct choice.