Area bounded by the parabola `y^(2)=2x` and the ordinates x=1, x=4 is

Options

A. `(4sqrt(2))/3` sq. units
B. `(28sqrt(2))/3` sq. units
C. `(56)/3` sq. units
D. `4/3` sq. units

Answer

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

### Step 1: Understand the equation of the parabola
The equation \( y^2 = 2x \) represents a parabola that opens to the right. To express \( y \) in terms of \( x \), we can take the square root:
\[ y = \sqrt{2x} \quad \text{and} \quad y = -\sqrt{2x} \]

### Step 2: Identify the area to be calculated
We need to calculate the area between the curves \( y = \sqrt{2x} \) and \( y = -\sqrt{2x} \) from \( x = 1 \) to \( x = 4 \). The area between these two curves can be calculated as:
\[ \text{Area} = 2 \int_{1}^{4} \sqrt{2x} \, dx \]
The factor of 2 accounts for the area above the x-axis and below the x-axis.

### Step 3: Set up the integral
Now we set up the integral:
\[ \text{Area} = 2 \int_{1}^{4} \sqrt{2x} \, dx \]

### Step 4: Simplify the integrand
We can simplify \( \sqrt{2x} \):
\[ \sqrt{2x} = \sqrt{2} \cdot \sqrt{x} \]
Thus, the integral becomes:
\[ \text{Area} = 2 \sqrt{2} \int_{1}^{4} \sqrt{x} \, dx \]

### Step 5: Integrate
The integral of \( \sqrt{x} \) is:
\[ \int \sqrt{x} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
Now we can evaluate the definite integral:
\[ \text{Area} = 2 \sqrt{2} \left[ \frac{2}{3} x^{3/2} \right]_{1}^{4} \]

### Step 6: Calculate the limits
Now we substitute the limits into the integral:
\[ = 2 \sqrt{2} \left( \frac{2}{3} (4^{3/2}) - \frac{2}{3} (1^{3/2}) \right) \]
Calculating \( 4^{3/2} \):
\[ 4^{3/2} = (2^2)^{3/2} = 2^3 = 8 \]
Thus, we have:
\[ = 2 \sqrt{2} \left( \frac{2}{3} (8) - \frac{2}{3} (1) \right) \]
\[ = 2 \sqrt{2} \left( \frac{16}{3} - \frac{2}{3} \right) \]
\[ = 2 \sqrt{2} \left( \frac{14}{3} \right) \]
\[ = \frac{28 \sqrt{2}}{3} \]

### Final Answer
The area bounded by the parabola \( y^2 = 2x \) and the ordinates \( x = 1 \) and \( x = 4 \) is:
\[ \text{Area} = \frac{28 \sqrt{2}}{3} \]