What is the area of the region bounded by the parabola $y^2 = 4x$ and the line $x = 1$?

Explanation

1. 01

   The region is symmetric about the $x$-axis; the upper half lies under $y = 2\sqrt{x}$ from 0 to 1.

2. 02

   $2\int_0^1 2\sqrt{x}\,dx = 2 \cdot \frac{4}{3} = \frac{8}{3}$.