A straight road runs east along the x-axis of a map…

Step 1. The crossings. $x^2 - 6x + 8 = (x - 2)(x - 4)$ is zero at $x = 2$ and $x = 4$. Both lie inside $[0, 5]$.

Step 2. The antiderivative. $F(x) = \dfrac{x^3}{3} - 3x^2 + 8x$. Then $F(0) = 0$, $F(2) = \dfrac{20}{3}$, $F(4) = \dfrac{16}{3}$ and $F(5) = \dfrac{20}{3}$.

Step 3. The integral. $F(5) - F(0) = \dfrac{20}{3} \approx 6.67$ square kilometres.

Step 4. The pieces. From $0$ to $2$ the value is $\dfrac{20}{3}$. From $2$ to $4$ it is $-\dfrac{4}{3}$, below the road. From $4$ to $5$ it is $\dfrac{4}{3}$.

Step 5. The total. $\dfrac{20}{3} + \dfrac{4}{3} + \dfrac{4}{3} = \dfrac{28}{3} \approx 9.33$ square kilometres.