Step 1: Understanding the Concept:
Area under a curve is typically calculated by integrating the function between its boundary points.
For a parabola \(y^2 = 4ax\), the latus rectum is a special line perpendicular to the axis of symmetry, passing through the focus.
The coordinates of the focus are \((a, 0)\), so the equation of the latus rectum is the vertical line \(x = a\).
Step 2: Key Formula or Approach:
1. Compare the given equation \(y^2 = 8x\) to the standard form \(y^2 = 4ax\).
2. Identify the value of \(a\) to find the boundary limit.
3. Total area = \(2 \times\) (Area of the upper half).
Step 3: Detailed Explanation:
Compare \(y^2 = 8x\) with \(y^2 = 4ax\):
\(4a = 8 \implies a = 2\).
The latus rectum line is \(x = 2\). The vertex is at \((0, 0)\). So, the limits of integration are from \(x = 0\) to \(x = 2\).
The equation for the upper half of the parabola is \(y = \sqrt{8x} = \sqrt{4 \times 2x} = 2\sqrt{2}\sqrt{x}\).
Set up the area integral:
\[ \text{Total Area} = 2 \int_{0}^{2} y \, dx = 2 \int_{0}^{2} 2\sqrt{2} x^{\frac{1}{2}} dx \]
\[ = 4\sqrt{2} \int_{0}^{2} x^{\frac{1}{2}} dx \]
Apply the power rule for integration (\(\int x^n dx = \frac{x^{n+1}}{n+1}\)):
\[ = 4\sqrt{2} \left[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right]_{0}^{2} = 4\sqrt{2} \cdot \frac{2}{3} \left[ x^{\frac{3}{2}} \right]_{0}^{2} \]
\[ = \frac{8\sqrt{2}}{3} (2^{\frac{3}{2}} - 0) \]
Note that \(2^{\frac{3}{2}} = \sqrt{2^3} = \sqrt{8} = 2\sqrt{2}\):
\[ = \frac{8\sqrt{2}}{3} \cdot 2\sqrt{2} = \frac{16 \cdot 2}{3} = \frac{32}{3} \]
Step 4: Final Answer:
The total bounded area is \(\frac{32}{3}\) square units, which is option (A).