The region bounded by y = 7/(1 + x^2), y = 0, x = 0 and x = 9 is rotated about the line x = 9. Using cylindrical shells, set up an integral for the volume of the resulting solid. Determine ONLY what the limits of integration are, and what the FUNCTION TO BE INTEGRATED is

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.

The region bounded by y = 7/(1 + x^2), y = 0, x = 0 and x = 9 is rotated about the line x = 9. Using cylindrical shells, set up an integral for the volume of the resulting solid. Determine ONLY what the limits of integration are, and what the FUNCTION TO BE INTEGRATED is.

Find the volume of a solid formed by rotating the region bounded by y = 7/(1 + x^2), y = 0, x = 0, and x = 9 around the line x = 9 using cylindrical shells. This problem involves setting up the limits of integration and determining the function to be integrated. Suitable for advanced high school and college calculus students.

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