se Washer Method to determine the volume of solid obtained by rotating
the region bounded by the following curves about the line x = −1:
y = −(x − 2)2 + 2, y = 1/x, x = 1, x = 3

The **Washer Method** is used to find the volume of a solid of revolution when the region between two curves is rotated about a line (other than the x-axis or y-axis). In this case, we are rotating the region bounded by the curves around the line \(x = -1\).

Given:
1. \( y = -(x - 2)^2 + 2 \)
2. \( y = \frac{1}{x} \)
3. \( x = 1 \)
4. \( x = 3 \)

The **Washer Method** formula for rotating about a vertical axis (in this case \(x = -1\)) is:

\[
V = \pi \int_{a}^{b} \left[ R_{\text{outer}}(x)^2 - R_{\text{inner}}(x)^2 \right] \, dx
\]

### Steps:

1. **Determine the bounds**: The region is bounded by \(x = 1\) and \(x = 3\), so the limits of integration are \(x = 1\) and \(x = 3\).

2. **Find the outer and inner radii**:
   - The outer radius is the distance from the line \(x = -1\) to the curve \(y = -(x - 2)^2 + 2\), which is:
   \[
   R_{\text{outer}}(x) = x - (-1) = x + 1
   \]
   
   - The inner radius is the distance from the line \(x = -1\) to the curve \(y = \frac{1}{x}\), which is:
   \[
   R_{\text{inner}}(x) = x - (-1) = x + 1
   \]

3. **Set up the integral**:
   Now we need to set up the integral for the volume using the formula:
   \[
   V = \pi \int_{1}^{3} \left[ (x + 1)^2 - (x + 1)^2 \right] \, dx
   \]
   However, since both the outer and inner radii are the same in this case, there is no difference between them, meaning that the volume of the solid is effectively zero.

### Conclusion:
In this particular case, the two curves are symmetric about the line \(x = -1\), resulting in the volume of the solid being zero.

Would you like any clarifications, or should I explore any specific details for you?

Here are some related questions for further exploration:
1. How would the problem change if the rotation was about the y-axis?
2. What if the rotation was about \( x = 1 \)?
3. How is the Washer Method different from the Shell Method?
4. What would happen if you rotated around the x-axis instead of \(x = -1\)?
5. How do you handle cases when the curves cross within the region?

**Tip**: Always sketch the region and axis of rotation to help visualize the radii when using the Washer Method.

Use Washer Method to determine the volume of solid obtained by rotating the region bounded by the following curves about the line x = −1: y = −(x − 2)² + 2, y = 1/x, x = 1, x = 3

This page provides a detailed solution to finding the volume of a solid of revolution using the Washer Method, with the region bounded by y = −(x − 2)² + 2, y = 1/x, x = 1, and x = 3, rotated around the line x = −1. Suitable for undergraduate calculus students.

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