What is the area of the region bounded by the parabola $y^{2} = 4x$ and the line $y = 2x - 4$?

Explanation

1. 01

   As $x$ in terms of $y$: the parabola is $x = \frac{y^{2}}{4}$ and the line $x = \frac{y + 4}{2}$. They meet where $y^{2} - 2y - 8 = 0$, at $y = -2$ and $y = 4$.

2. 02

   Horizontal strips run from the parabola to the line: $\int_{-2}^{4} \left(\frac{y + 4}{2} - \frac{y^{2}}{4}\right)dy$.

3. 03

   $\left[\frac{y^{2}}{4} + 2y - \frac{y^{3}}{12}\right]_{-2}^{4} = \frac{20}{3} + \frac{7}{3} = 9$ square units.