Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = x^(1/2), y = 0, x = 4; about x = 9

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves \( y = \sqrt{x} \), \( y = 0 \), and \( x = 4 \) about the line \( x = 9 \), we can use the **method of cylindrical shells**. Here's how the setup proceeds step by step:

### Step 1: Identify the bounds of the region
The region is bounded by:
- \( y = \sqrt{x} \) (the curve)
- \( y = 0 \) (the x-axis)
- \( x = 4 \) (vertical line at \( x = 4 \))

Thus, \( x \) varies from 0 to 4.

### Step 2: Formula for volume using cylindrical shells
The formula for the volume of a solid of revolution using cylindrical shells is:

\[
V = 2\pi \int_{a}^{b} \left(\text{radius of the shell}\right) \left(\text{height of the shell}\right) \, dx
\]

### Step 3: Radius and height of the shell
- **Radius**: The radius of a shell is the horizontal distance from the line \( x = 9 \) to the point at \( x \). Thus, the radius is \( 9 - x \).
- **Height**: The height of a shell is the vertical distance from the curve \( y = \sqrt{x} \) to the x-axis, which is simply \( y = \sqrt{x} \).

### Step 4: Set up the integral
Now, the volume is:

\[
V = 2\pi \int_{0}^{4} (9 - x)(\sqrt{x}) \, dx
\]

This is the integral that represents the volume of the solid.

Would you like any more details on the steps or explanation?

Here are 5 relative questions:
1. How does the method of disks differ from cylindrical shells in this context?
2. What is the geometric interpretation of the radius and height in the shell method?
3. Can you visualize the solid formed by rotating this region about \( x = 9 \)?
4. How would the setup change if the region was rotated about the y-axis?
5. What if the region was bounded by \( y = x^2 \) instead of \( y = \sqrt{x} \)?

**Tip**: For cylindrical shells, the radius is always the distance from the axis of rotation to the variable of integration.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x^(1/2), y = 0, x = 4; about x = 9

Learn how to set up an integral for the volume of a solid obtained by rotating the region bounded by y = √x, y = 0, and x = 4 about the line x = 9. This problem involves the method of cylindrical shells and provides a step-by-step setup process, suitable for high school and college calculus students.

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