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

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

The shaded region lies under the curve \(y = \sqrt{x}\), above the x-axis, between \(x = 1\) and \(x = 4\). Its area is the definite integral:

    \( \displaystyle A = \int_{1}^{4}\sqrt{x}\,dx = \left[\frac{2}{3}x^{3/2}\right]_{1}^{4} \)

    \( \displaystyle = \frac{2}{3}\left(4^{3/2} - 1^{3/2}\right) = \frac{2}{3}(8 - 1) \)

    \( \displaystyle = \frac{14}{3} \text{ square units} \)

Note: It seems that there is a mistake in the original NTA paper, which asks for “the nearest integral value” of the shaded area when it should ask for the shaded area itself. The area is \(\frac{14}{3}\), about 4.67, whose nearest integer, 5, is not among the options. The question can still be solved because Option (3) gives the exact value of the area.