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 between the curves $y = f(x)$ and $y = g(x)$, we must account for the areas above and below the x-axis.

1. Identify the intervals from the graph where $f(x)$ is above $g(x)$ and where $g(x)$ is above $f(x)$. Here, we observe that the curves intersect at several points, impacting the areas positively and negatively.

2. For the interval $[-4, -3]$, $f(x)$ is above $g(x)$, hence we calculate this area as $\int_{-4}^{-3} f(x) - g(x) \,dx$.

3. For the interval $[-3, -1]$, $g(x)$ is above $f(x)$, so the relevant area is $\int_{-3}^{-1} g(x) - f(x) \,dx$.

4. Lastly, for the interval $[-1, 4]$, $f(x)$ is again above $g(x)$, contributing an area of $\int_{-1}^{4} f(x) - g(x) \,dx$.

Combining these, we find that the sum of the areas above and below sums to provide the total shaded area as follows:

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

Thus, the correct answer is option D.