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 $x = 9$.

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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.