To find the volume of the solid formed by rotating the region bounded by $y = \frac{7}{1 + x^2}$, $y = 0$, $x = 0$, and $x = 9$ around the line $x = 9$ using the method of cylindrical shells, follow these steps:

Step 1: Formula for the Volume Using Cylindrical Shells

The formula for the volume using cylindrical shells is:

$V = \int_{a}^{b} 2\pi ( \text{radius} ) ( \text{height} ) \, dx$

Step 2: Identify the Radius and Height

• The radius of a cylindrical shell is the distance from any point $x$ in the region to the axis of rotation $x = 9$. This is given by: $\text{radius} = 9 - x$

• The height of a cylindrical shell is the function that defines the region being rotated, which is $y = \frac{7}{1 + x^2}$.

Thus, the function to be integrated is: $2\pi (9 - x) \left( \frac{7}{1 + x^2} \right)$

Step 3: Limits of Integration

The limits of integration correspond to the $x$-values that bound the region. These are from $x = 0$ to $x = 9$.

Final Answer:

• Limits of integration: $x = 0$ to $x = 9$
• Function to be integrated: $2\pi (9 - x) \left( \frac{7}{1 + x^2} \right)$

Would you like a more detailed explanation, or have any questions?

Here are 5 related questions to consider:

1. How does the method of cylindrical shells compare to the method of disks/washers for volume computation?
2. What would change if the region was rotated around the y-axis instead?
3. How do the limits of integration change if the region is rotated around a different line, say $x = 10$?
4. How would you set up the integral if the region was rotated about the line $x = -1$?
5. How can we calculate the exact volume after setting up the integral?

Tip: The radius in cylindrical shell method is always the distance from the axis of rotation to the shell. Ensure the radius is accurately determined based on the rotation axis.