Step-by-Step Solution

1. The curve is given by \(y^2 = x\), which means \(y = \sqrt{x}\) (since we are considering the area bounded by the x-axis, we take the positive square root).<\/li>
2. The region is bounded by \(y = \sqrt{x}\), \(x = 4\), and the x-axis.<\/li>
3. The area of the shaded region can be found by integrating the function \(y = \sqrt{x}\) with respect to \(x\) from 0 to 4.<\/li>
4. So, the area is given by \( \int_{0}^{4} \sqrt{x} \, dx\).<\/li>
5. However, since the region is symmetric about the x-axis, we can also express the area as twice the integral from 0 to 2 with respect to y. Since \(y^2 = x\), when \(x = 4\), \(y = 2\). The area can be expressed as \(2 \int_{0}^{2} (4 - y^2) \, dy\). This is not one of the options.<\/li>
6. The integral \( \int_{0}^{4} \sqrt{x} \, dx\) represents the area under the curve \(y = \sqrt{x}\) from \(x = 0\) to \(x = 4\). This is the area of the region bounded by the curve, the line \(x = 4\), and the x-axis.<\/li>
7. The integral \(2 \int_{0}^{4} \sqrt{x} \, dx\) would represent twice the area, which is not what we want.<\/li>