The area of the parabola `y ^(2) = 4ax` bounded by the latusrectum is `"_________".`

Answer

To find the area of the parabola \( y^2 = 4ax \) bounded by the latus rectum, we can follow these steps:

### Step 1: Understand the parabola and its latus rectum
The given parabola \( y^2 = 4ax \) opens to the right, with its vertex at the origin (0,0) and focus at the point \( (a, 0) \). The latus rectum is a line segment perpendicular to the axis of symmetry of the parabola (the x-axis in this case) that passes through the focus. The endpoints of the latus rectum are \( (a, 2a) \) and \( (a, -2a) \).

### Step 2: Set up the area calculation
The area bounded by the parabola and the latus rectum can be calculated by finding the area under the curve from \( x = 0 \) to \( x = a \) and then doubling it because the parabola is symmetric about the x-axis.

### Step 3: Express \( y \) in terms of \( x \)
From the equation of the parabola \( y^2 = 4ax \), we can express \( y \) as:
\[ y = \sqrt{4ax} = 2\sqrt{ax} \]
We will consider the positive root since we are calculating the area above the x-axis.

### Step 4: Set up the integral for the area
The area \( A \) under the curve from \( x = 0 \) to \( x = a \) is given by:
\[ A = \int_0^a y \, dx = \int_0^a 2\sqrt{ax} \, dx \]

### Step 5: Calculate the integral
First, factor out the constant \( 2\sqrt{a} \):
\[ A = 2\sqrt{a} \int_0^a \sqrt{x} \, dx \]
Now, we need to compute the integral \( \int_0^a \sqrt{x} \, dx \):
\[ \int \sqrt{x} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
Evaluating this from \( 0 \) to \( a \):
\[ \int_0^a \sqrt{x} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_0^a = \frac{2}{3} a^{3/2} \]

### Step 6: Substitute back into the area formula
Now substituting this back into our area formula:
\[ A = 2\sqrt{a} \cdot \frac{2}{3} a^{3/2} = \frac{4}{3} a^2 \]

### Step 7: Double the area for symmetry
Since we only calculated the area above the x-axis, we need to double it to account for the area below the x-axis:
\[ \text{Total Area} = 2A = 2 \cdot \frac{4}{3} a^2 = \frac{8}{3} a^2 \]

### Final Answer
Thus, the area of the parabola \( y^2 = 4ax \) bounded by the latus rectum is:
\[ \frac{8}{3} a^2 \]