Area of the portion of the parabola `y^(2)=4ax` included between the X-axis, the ordinate at x = 2a and its latus rectum is

Options

A. `(a^(2)(2sqrt(2)-1))/(3)`
B. `(4a^(2)(2sqrt(2)-1))/(3)`
C. `(8a^(2)(2sqrt(2)-1))/(3)`
D. None of these

Answer

To find the area of the portion of the parabola \( y^2 = 4ax \) included between the X-axis, the ordinate at \( x = 2a \), and its latus rectum, we can follow these steps:

### Step 1: Understand the Parabola and its Latus Rectum
The equation of the parabola is given by \( y^2 = 4ax \). The focus of this parabola is at the point \( (a, 0) \). The latus rectum of the parabola is a vertical line that passes through the focus, which can be represented by the equation \( x = a \).

### Step 2: Identify the Area to be Calculated
We need to find the area bounded by:
- The X-axis (where \( y = 0 \))
- The vertical line \( x = 2a \)
- The latus rectum (the line \( x = a \))
- The parabola itself

### Step 3: Set Up the Integral
The area under the curve from \( x = a \) to \( x = 2a \) can be calculated using the integral of the function \( y = 2\sqrt{ax} \) (derived from \( y^2 = 4ax \)).

The area \( A \) can be represented as:
\[ A = \int_{a}^{2a} 2\sqrt{ax} \, dx \]

### Step 4: Simplify the Integral
We can factor out the constant \( 2\sqrt{a} \):
\[ A = 2\sqrt{a} \int_{a}^{2a} \sqrt{x} \, dx \]

### Step 5: Compute the Integral
The integral \( \int \sqrt{x} \, dx \) can be computed as follows:
\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} \]

Now we can evaluate the definite integral:
\[ A = 2\sqrt{a} \left[ \frac{2}{3} x^{3/2} \right]_{a}^{2a} \]

### Step 6: Evaluate the Limits
Calculating the upper limit:
\[ \frac{2}{3} (2a)^{3/2} = \frac{2}{3} \cdot 2^{3/2} \cdot a^{3/2} = \frac{2 \cdot 2\sqrt{2}}{3} a^{3/2} = \frac{4\sqrt{2}}{3} a^{3/2} \]

Calculating the lower limit:
\[ \frac{2}{3} (a)^{3/2} = \frac{2}{3} a^{3/2} \]

### Step 7: Combine the Results
Now substituting back into the area expression:
\[ A = 2\sqrt{a} \left( \frac{4\sqrt{2}}{3} a^{3/2} - \frac{2}{3} a^{3/2} \right) \]
\[ A = 2\sqrt{a} \left( \frac{4\sqrt{2} - 2}{3} a^{3/2} \right) \]
\[ A = \frac{2\sqrt{a} a^{3/2}}{3} (4\sqrt{2} - 2) = \frac{2a^2}{3} (4\sqrt{2} - 2) \]

### Final Result
Thus, the area of the portion of the parabola is:
\[ A = \frac{2a^2}{3} (4\sqrt{2} - 2) \]