Two cycle lanes are drawn on a map in kilometres…

Step 1. Meeting points. $x^3 - 2x = x$ gives $x(x^2 - 3) = 0$, so $x = -\sqrt{3}$, $0$ and $\sqrt{3}$. The largest is $\sqrt{3} \approx 1.73$.

Step 2. Which is above? On $(0, \sqrt{3})$ take $x = 1$. The line gives $1$ and the cubic gives $-1$. The line is above. On $(-\sqrt{3}, 0)$ the cubic is above.

Step 3. One patch. $\displaystyle\int_0^{\sqrt{3}} (3x - x^3)\,dx = \dfrac{3 \times 3}{2} - \dfrac{9}{4} = \dfrac{9}{4}$.

Step 4. Both patches. The patches are equal, so the total area is $2 \times \dfrac{9}{4} = \dfrac{9}{2}$.