Find the area of region bounded by the line $x = 2$ and the parabola ${y^2} = 8x$

We have, line $x = 2$ and the parabola ${y^2} = 8x$
Graphs of parabola and line are as shown in the following figure.

From the figure, area of shaded region

$A = 2\int_0^2 {\sqrt {8x} } dx$
$= 4\sqrt 2 \int_0^{1/2} d x = 4\sqrt 2 \left[ {\frac{2}{3}{x^{3/2}}} \right]_0^2$

$= 4\sqrt 2 \left[ {\frac{2}{3} \cdot 2\sqrt 2 - 0} \right] = \frac{{32}}{3}$sq. units