The area of the region bounded by parabola `y^(2)=x` and the straight line `2y = x` is

Options

A. `4/3"sq units"`
B. `1" sq unit "`
C. `2/3 " sq unit"`
D. `1/3" sq units"`

Explanation

To find the area of the region bounded by the parabola \( y^2 = x \) and the straight line \( 2y = x \), we can follow these steps:

### Step 1: Find the points of intersection
We need to find the points where the parabola and the line intersect.

1. The equations are:
\[ y^2 = x \quad \text{(1)} \]
\[ 2y = x \quad \text{(2)} \]

2. Substitute equation (2) into equation (1):
\[ y^2 = 2y \]

3. Rearranging gives:
\[ y^2 - 2y = 0 \]

4. Factoring out \( y \):
\[ y(y - 2) = 0 \]
This gives us \( y = 0 \) and \( y = 2 \).

5. Now, find the corresponding \( x \) values using equation (2):
- For \( y = 0 \):
\[ x = 2(0) = 0 \quad \Rightarrow \quad (0, 0) \]
- For \( y = 2 \):
\[ x = 2(2) = 4 \quad \Rightarrow \quad (4, 2) \]

### Step 2: Set up the integral for the area
The area \( A \) between the curves from \( x = 0 \) to \( x = 4 \) can be found by integrating the difference between the upper curve and the lower curve.

1. The upper curve is the parabola \( y = \sqrt{x} \) and the lower curve is the line \( y = \frac{x}{2} \).

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

### Step 3: Evaluate the integral
1. Break down the integral:
\[ A = \int_{0}^{4} \sqrt{x} \, dx - \int_{0}^{4} \frac{x}{2} \, dx \]

2. Calculate the first integral:
\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} \quad \text{(from the power rule)} \]
Evaluating from 0 to 4:
\[ \left[ \frac{2}{3} x^{3/2} \right]_{0}^{4} = \frac{2}{3} (4^{3/2}) - \frac{2}{3} (0^{3/2}) = \frac{2}{3} (8) = \frac{16}{3} \]

3. Calculate the second integral:
\[ \int \frac{x}{2} \, dx = \frac{1}{4} x^2 \]
Evaluating from 0 to 4:
\[ \left[ \frac{1}{4} x^2 \right]_{0}^{4} = \frac{1}{4} (4^2) - \frac{1}{4} (0^2) = \frac{1}{4} (16) = 4 \]

4. Combine the results:
\[ A = \frac{16}{3} - 4 = \frac{16}{3} - \frac{12}{3} = \frac{4}{3} \]

### Final Answer
The area of the region bounded by the parabola \( y^2 = x \) and the straight line \( 2y = x \) is:
\[ \boxed{\frac{4}{3}} \text{ square units} \]