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 about the line \(x = 9\), we can use the **method of cylindrical shells**. This method is appropriate when revolving around a vertical axis (not necessarily \(x = 0\)).

### Given:
- Curves: \(y = \sqrt{x}\), \(y = 0\), and \(x = 4\)
- Axis of rotation: \(x = 9\)

### Steps to Set Up the Integral:

1. **Set up the radius of each shell**:  
   The radius of each shell is the horizontal distance from a point on the curve \(x\) to the line \(x = 9\). Therefore, the radius is \(9 - x\).

2. **Set up the height of each shell**:  
   The height of each shell is given by the function \(y = \sqrt{x}\), as this is the distance from the curve to \(y = 0\).

3. **Set up the differential volume of a cylindrical shell**:  
   The formula for the volume of a cylindrical shell is given by:
   \[
   \text{Volume of shell} = 2\pi (\text{radius})(\text{height})(\text{thickness})
   \]
   In our case:
   \[
   dV = 2\pi (9 - x)(\sqrt{x}) \, dx
   \]

4. **Determine the bounds of integration**:  
   The region is bounded by \(x = 0\) and \(x = 4\), so these are the limits of integration.

5. **Set up the integral**:  
   The integral that represents the volume is:
   \[
   V = \int_{0}^{4} 2\pi (9 - x)(\sqrt{x}) \, dx
   \]

### Final Integral (Not evaluated):
\[
V = 2\pi \int_{0}^{4} (9 - x)\sqrt{x} \, dx
\]

This is the integral that represents the volume of the solid obtained by rotating the given region about the line \(x = 9\).

---

Would you like more details on this setup or any specific part of the process?

Here are 5 related questions to consider:

1. How would the integral change if we rotated about a different line, such as \(x = 0\)?
2. What method would be used if we rotated about a horizontal axis, like \(y = 0\)?
3. How does the cylindrical shells method differ from the disk/washer method?
4. How can we visualize the shape of the solid formed by this rotation?
5. What does the integrand \((9 - x)\sqrt{x}\) represent geometrically?

**Tip**: When setting up integrals for volumes of revolution, always clearly distinguish between the radius and height of the shells or disks based on the method you choose.

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 formed by rotating the region bounded by y = √x, y = 0, and x = 4 around the line x = 9 using the method of cylindrical shells. This problem explores key calculus concepts like volumes of revolution and integral calculus, suitable for college-level students.

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