15. Find the volume of the solid generated by revolving the region bounded by y = 2
p
x and the lines

y = 2, x = 4, about the line y = 2.

Solution: Volume is 14π
3

To find the volume of the solid generated by revolving the region bounded by \( y = 2\sqrt{x} \), the lines \( y = 2 \) and \( x = 4 \), about the line \( y = 2 \), we can use the **method of cylindrical shells**.

Here is the process step by step:

### Step 1: Understanding the region and bounds
- The curve is \( y = 2\sqrt{x} \), which implies \( x = \frac{y^2}{4} \).
- The line \( y = 2 \) is a horizontal line, and the line \( x = 4 \) is vertical. We are revolving the region about the line \( y = 2 \).
- The boundaries of the region are:
  - For \( y = 2 \), it corresponds to \( x = 1 \) on the curve \( y = 2\sqrt{x} \), but the vertical boundary at \( x = 4 \) will serve as our upper bound for \( x \).

### Step 2: Setting up the integral (Cylindrical Shell Method)
- The distance from the axis of revolution (which is the line \( y = 2 \)) is given by \( |2 - y| \), where \( y = 2\sqrt{x} \).
- The height of the shell is the function \( 2\sqrt{x} - 2 \), and we will use the volume of cylindrical shells formula:

\[
V = \int_{x_{\text{min}}}^{x_{\text{max}}} 2\pi \cdot \text{radius} \cdot \text{height} \, dx
\]

The height of each shell is the function \( 2\sqrt{x} \), and the radius will be \( x \), and integrating from 0 to 2.

### Step 

Find the volume of the solid generated by revolving the region bounded by y = 2√x and the lines y = 2, x = 4, about the line y = 2.

Learn how to calculate the volume of the solid formed by revolving the region bounded by the curve y = 2√x, the line y = 2, and x = 4, around the line y = 2. This solution uses the cylindrical shell method and covers concepts of integration and volume of solids. Suitable for undergraduate calculus students.

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