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The image shows a problem where we need to find the volume of the solid generated by revolving the shaded region around the x-axis using the shell method. The region is bounded by two curves: \( x = 3y^2 \) and \( y = \sqrt{3} \).

### Shell Method Formula for Volume:
The general formula for volume using the shell method when revolving around the x-axis is:
\[
V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} (\text{radius}) \cdot (\text{height}) \, dy
\]
In this case:
- The radius of a shell at a given \( y \) is \( y \) (distance from the x-axis).
- The height of the shell at a given \( y \) is the horizontal distance between the two curves, i.e., the difference between the rightmost curve \( x = 9 \) and the leftmost curve \( x = 3y^2 \).

### Setting up the integral:
1. **Limits of integration**: From the graph, \( y_{\text{min}} = 0 \) and \( y_{\text{max}} = \sqrt{3} \).
2. **Height of the shell**: The height is given by the difference in x-values, which is \( 9 - 3y^2 \).

Thus, the integral becomes:
\[
V = 2\pi \int_0^{\sqrt{3}} y(9 - 3y^2) \, dy
\]

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

Would you like to proceed with the computation or further explanation? 

Here are 5 related questions:
1. What happens if we rotate around the y-axis instead of the x-axis?
2. How would the setup change if we used the disk method instead of the shell method?
3. What is the significance of the bounds \( y = 0 \) and \( y = \sqrt{3} \)?
4. How do we interpret the geometry of the problem in 3D?
5. Can we apply the same method for solids with more complex boundaries?

**Tip**: The shell method is particularly useful when the region is easier to describe in terms of vertical slices (along the y-axis), as in this problem.

Use the shell method to find the volume of the solid generated by revolving the shaded region about the x-axis.

Learn how to use the shell method to calculate the volume of a solid generated by revolving a region bounded by curves around the x-axis. This problem involves calculus concepts like setting up integrals and solving for volumes of solids.

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