The volume of the solid formed by rotating the area enclosed between the curve `y^(2)=4x,x=4` and x = 5 about X-axis is (in cubic units)

Options

A. `18pi`
B. `36pi`
C. `9pi`
D. `24pi`

Explanation

To find the volume of the solid formed by rotating the area enclosed between the curve \( y^2 = 4x \), \( x = 4 \), and \( x = 5 \) about the x-axis, we can use the method of disks. Here’s a step-by-step solution:

### Step 1: Identify the curve and limits of integration
The given curve is \( y^2 = 4x \). We also have the vertical lines \( x = 4 \) and \( x = 5 \). The area we are interested in is between these two vertical lines.

### Step 2: Express \( y \) in terms of \( x \)
From the equation \( y^2 = 4x \), we can express \( y \) as:
\[ y = \sqrt{4x} = 2\sqrt{x} \]
This is valid since we are considering the positive root for the area above the x-axis.

### Step 3: Set up the volume integral using the disk method
The volume \( V \) of the solid of revolution when rotating around the x-axis is given by the integral:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
In our case, \( f(x) = y = 2\sqrt{x} \), and the limits of integration are \( a = 4 \) and \( b = 5 \). Thus, we have:
\[ V = \pi \int_{4}^{5} (2\sqrt{x})^2 \, dx \]

### Step 4: Simplify the integrand
Calculating \( (2\sqrt{x})^2 \):
\[ (2\sqrt{x})^2 = 4x \]
So, the volume integral becomes:
\[ V = \pi \int_{4}^{5} 4x \, dx \]

### Step 5: Factor out the constant
We can factor out the constant \( 4 \):
\[ V = 4\pi \int_{4}^{5} x \, dx \]

### Step 6: Evaluate the integral
Now we compute the integral:
\[ \int x \, dx = \frac{x^2}{2} \]
Thus, we evaluate:
\[ \int_{4}^{5} x \, dx = \left[ \frac{x^2}{2} \right]_{4}^{5} = \frac{5^2}{2} - \frac{4^2}{2} = \frac{25}{2} - \frac{16}{2} = \frac{9}{2} \]

### Step 7: Substitute back to find the volume
Now substitute this result back into the volume formula:
\[ V = 4\pi \cdot \frac{9}{2} = 18\pi \]

### Final Answer
Thus, the volume of the solid formed by rotating the area about the x-axis is:
\[ \boxed{18\pi} \text{ cubic units} \]

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