Step 1: Understanding the Question:
We need to find the area of the region enclosed by the parabola \( y^2 = 4x \), the line \( x = 3 \), and implicitly bounded by the x-axis from \(x=0\) to \(x=3\). The parabola is symmetric about the x-axis.
Step 2: Key Formula or Approach:
The area of a region bounded by a curve \( y = f(x) \), the x-axis, and the lines \( x = a \) and \( x = b \) is given by \( A = \int_{a}^{b} f(x) \, dx \).
Since the parabola is symmetric about the x-axis, we can calculate the area of the upper half and multiply it by 2.
From \( y^2 = 4x \), the upper half is represented by \( y = \sqrt{4x} = 2\sqrt{x} \).
The limits of integration are from \( x=0 \) to \( x=3 \).
Total Area \( = 2 \times \int_{0}^{3} 2\sqrt{x} \, dx \).
Step 3: Detailed Explanation:
The integral to be calculated is:
\[ A = 4 \int_{0}^{3} x^{1/2} \, dx \]
Integrate \( x^{1/2} \):
\[ A = 4 \left[ \frac{x^{1/2 + 1}}{1/2 + 1} \right]_{0}^{3} = 4 \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{3} = 4 \left[ \frac{2}{3}x^{3/2} \right]_{0}^{3} \]
Now, evaluate the definite integral:
\[ A = \frac{8}{3} [3^{3/2} - 0^{3/2}] \]
\[ A = \frac{8}{3} [ (3\sqrt{3}) - 0 ] \]
\[ A = \frac{8}{3} \times 3\sqrt{3} \]
\[ A = 8\sqrt{3} \]
Step 4: Final Answer:
The area of the region is \( 8\sqrt{3} \) square units.