The diagram shows the graphs of the functions $f(x)$ and $g(x)$ - HSC - SSCE Mathematics Extension 1 - Question 4 - 2023 - Paper 1

To find the area between the curves from $x = a$ to $x = c$, we need to evaluate the integral from $a$ to $c$ of the difference between the functions:

$\text{Area} = \int_{a}^{c} (f(x) - g(x)) \, dx$

Using the given integrals, we can break this down as follows:

$\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx$

We know:

• From $a$ to $c$, $\int_{a}^{c} f(x) \, dx = 10$.
• From $a$ to $b$, we can represent it as $x$, then:
  • knowing $g(x)$ over the interval $[a, b]$, we can use the previously calculated integrals:
  $\int_{b}^{c} g(x) \, dx = 3$
• Now, we know $\int_{a}^{b} g(x) \, dx = -2$, which leads us to $\int_{a}^{c} g(x) \, dx = -2 + 3 = 1.$

Thus, combining:

$\int_{a}^{c} (f(x) - g(x)) \, dx = \int_{a}^{c} f(x) \, dx - \int_{a}^{c} g(x) \, dx$

This yields:

$\text{Area} = 10 - 1 = 9.$

So, the area between the curves is 9.