The nearest integral value of the shaded area shown below is:

$\frac{14}{3}$ square units

The correct answer is Option (3) → $\frac{14}{3}$ square units

The area under the curve $y = \sqrt{x}$ from $x = 1$ to $x = 4$ is given by:

$A = \int_{1}^{4} \sqrt{x} \, dx$

Using the power rule for integration:

$A = \int_{1}^{4} x^{\frac{1}{2}} \, dx = \frac{2}{3} x^{\frac{3}{2}} \bigg|_{1}^{4}$

$= \frac{2}{3} \left( 4^{\frac{3}{2}} - 1^{\frac{3}{2}} \right)$

$= \frac{2}{3} \left( 8 - 1 \right) = \frac{2}{3} \times 7 = \frac{14}{3} \approx 4.67$

The nearest integer value is $5$.