What is the area of the parabola `y^(2) = x` bounde by its latus rectum ?

Options

A. `1/12` square unit
B. `1/6` square unit
C. `1/3` square unit
D. None of the above

Explanation

To find the area of the parabola \( y^2 = x \) bounded by its latus rectum, we will follow these steps:

### Step 1: Understand the parabola and its latus rectum
The given parabola is \( y^2 = x \). The latus rectum of a parabola in the form \( y^2 = 4ax \) is a vertical line segment that passes through the focus of the parabola. For the parabola \( y^2 = x \), we can rewrite it as \( y^2 = 4 \cdot \frac{1}{4} x \), which indicates that \( a = \frac{1}{4} \).

### Step 2: Identify the focus and endpoints of the latus rectum
The focus of the parabola \( y^2 = x \) is at the point \( (a, 0) = \left(\frac{1}{4}, 0\right) \). The endpoints of the latus rectum can be found at \( \left(\frac{1}{4}, 2a\right) \) and \( \left(\frac{1}{4}, -2a\right) \). Since \( a = \frac{1}{4} \), the endpoints are:
- \( \left(\frac{1}{4}, \frac{1}{2}\right) \)
- \( \left(\frac{1}{4}, -\frac{1}{2}\right) \)

### Step 3: Set up the integral for area calculation
The area bounded by the parabola and the latus rectum can be calculated by integrating the function \( y = \sqrt{x} \) from \( x = 0 \) to \( x = \frac{1}{4} \). Since the parabola is symmetric about the x-axis, we can calculate the area above the x-axis and then double it.

The area \( A \) can be expressed as:
\[ A = 2 \int_{0}^{\frac{1}{4}} \sqrt{x} \, dx \]

### Step 4: Calculate the integral
Now we will compute the integral:
\[ \int \sqrt{x} \, dx = \int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} x^{\frac{3}{2}} \]
Now we evaluate this from \( 0 \) to \( \frac{1}{4} \):
\[ A = 2 \left[ \frac{2}{3} x^{\frac{3}{2}} \right]_{0}^{\frac{1}{4}} = 2 \left[ \frac{2}{3} \left(\frac{1}{4}\right)^{\frac{3}{2}} - 0 \right] \]
Calculating \( \left(\frac{1}{4}\right)^{\frac{3}{2}} \):
\[ \left(\frac{1}{4}\right)^{\frac{3}{2}} = \frac{1}{8} \]
So, substituting back:
\[ A = 2 \left[ \frac{2}{3} \cdot \frac{1}{8} \right] = 2 \cdot \frac{2}{24} = \frac{1}{6} \]

### Final Answer
Thus, the area of the parabola \( y^2 = x \) bounded by its latus rectum is:
\[ \boxed{\frac{1}{6}} \text{ square units.} \]