Find the area of the region bounded by the parabola ` y^ ( 2 ) = 4x`, X- axis and the lines ` x = 1, x = 4`.

Answer

To find the area of the region bounded by the parabola \( y^2 = 4x \), the x-axis, and the lines \( x = 1 \) and \( x = 4 \), we can follow these steps:

### Step 1: Understand the given parabola
The equation of the parabola is given by \( y^2 = 4x \). This can be rewritten to express \( y \) in terms of \( x \):
\[ y = 2\sqrt{x} \quad \text{and} \quad y = -2\sqrt{x} \]
This means the parabola opens to the right and is symmetric about the x-axis.

### Step 2: Identify the area to be calculated
We need to find the area between the parabola, the x-axis, and the vertical lines \( x = 1 \) and \( x = 4 \). Since the area above the x-axis is the same as the area below it (due to symmetry), we can calculate the area above the x-axis and then double it.

### Step 3: Set up the integral
The area \( A \) can be expressed as:
\[ A = 2 \int_{1}^{4} y \, dx \]
Substituting \( y = 2\sqrt{x} \):
\[ A = 2 \int_{1}^{4} 2\sqrt{x} \, dx = 4 \int_{1}^{4} \sqrt{x} \, dx \]

### Step 4: Calculate the integral
Now we compute the integral \( \int \sqrt{x} \, dx \):
\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} + C \]
Thus, we evaluate:
\[ A = 4 \left[ \frac{2}{3} x^{3/2} \right]_{1}^{4} \]
Calculating the definite integral:
\[ = 4 \left( \frac{2}{3} (4^{3/2}) - \frac{2}{3} (1^{3/2}) \right) \]
Calculating \( 4^{3/2} = 8 \) and \( 1^{3/2} = 1 \):
\[ = 4 \left( \frac{2}{3} (8) - \frac{2}{3} (1) \right) = 4 \left( \frac{16}{3} - \frac{2}{3} \right) = 4 \left( \frac{14}{3} \right) = \frac{56}{3} \]

### Step 5: Final result
Thus, the area of the region bounded by the parabola, the x-axis, and the lines \( x = 1 \) and \( x = 4 \) is:
\[ \boxed{\frac{56}{3}} \text{ square units} \]