Step 1: Concept Identification:
The objective is to determine the area of a region bounded by a parabola and a line. This is a standard definite integration problem. The parabola is defined by \(y^2 = 4x\), which opens to the right and has its vertex at the origin.
Step 2: Method Selection:
The area bounded by a curve \(y = f(x)\), the x-axis, and vertical lines \(x = a\) and \(x = b\) is calculated using the definite integral \(\int_{a}^{b} f(x) dx\). Given the symmetry of the parabola about the x-axis, the total area can be found by calculating the area in the first quadrant and then doubling it.
Step 3: Calculation Procedure:
For the parabola \(y^2 = 4x\), the upper half in the first quadrant is represented by \(y = \sqrt{4x} = 2\sqrt{x}\).
The region is defined by the x-axis (implicitly) and the vertical lines \(x = 0\) (the y-axis) and \(x = 1\).
The area in the first quadrant (\(A_1\)) is computed as:
\[ A_1 = \int_{0}^{1} y \, dx = \int_{0}^{1} 2\sqrt{x} \, dx \]
\[ A_1 = 2 \int_{0}^{1} x^{1/2} \, dx \]
Applying the power rule of integration:
\[ A_1 = 2 \left[ \frac{x^{1/2 + 1}}{1/2 + 1} \right]_{0}^{1} = 2 \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{1} = 2 \left[ \frac{2}{3}x^{3/2} \right]_{0}^{1} \]
\[ A_1 = \frac{4}{3} [x^{3/2}]_{0}^{1} = \frac{4}{3} (1^{3/2} - 0^{3/2}) = \frac{4}{3}(1 - 0) = \frac{4}{3} \]
This \(A_1\) represents the area above the x-axis. Due to symmetry, the area below the x-axis is also \(\frac{4}{3}\).
The total area is the sum of these two parts:
\[ A_{\text{total}} = 2 \times A_1 = 2 \times \frac{4}{3} = \frac{8}{3} \]
Step 4: Conclusion:
The total area of the region is $\frac{8{3}$} square units.