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

$\frac{32}{3}$ square units

The correct answer is Option (2) → $\frac{32}{3}$ square units

We have, $y^2 = 8x$ and $x = 2$.

$∴$ Area of shaded region $= 2 \int\limits_{0}^{2} \sqrt{8x} \, dx = 2 \cdot 2\sqrt{2} \int\limits_{0}^{2} x^{1/2} \, dx$

$= 4 \cdot \sqrt{2} \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{2} = 4\sqrt{2} \left[ \frac{2}{3} \cdot 2\sqrt{2} - 0 \right]$

$= \frac{32}{3} \text{ sq. units}$