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The problem asks us to find the volume of the solid \( S \) obtained by rotating the region \( R \), which is bounded by the curves \( y = \ln(x) \), \( y = 1 \), and \( x = 1 \), about the y-axis.

### Steps to solve:

1. **Identify the region of integration**:
   - The region \( R \) is bounded by \( y = \ln(x) \), \( y = 1 \), and \( x = 1 \). Since the curve is rotated around the **y-axis**, we need to express \( x \) in terms of \( y \) from the equation \( y = \ln(x) \), which gives:
     \[
     x = e^y.
     \]

2. **Volume using the shell method**:
   - To compute the volume of revolution around the y-axis, the method of cylindrical shells is appropriate. The volume \( V \) is given by:
     \[
     V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} \left( \text{radius} \cdot \text{height} \right) \, dy,
     \]
     where the radius is \( x = e^y \) and the height is \( x = e^y - 0 \).

3. **Set up the integral**:
   - The limits of integration come from the bounds \( y = \ln(x) \) and \( y = 1 \). For the given problem, \( y_{\text{min}} = 0 \) (since \( \ln(1) = 0 \)) and \( y_{\text{max}} = 1 \). The integral becomes:
     \[
     V = 2\pi \int_{0}^{1} e^y \cdot e^y \, dy = 2\pi \int_{0}^{1} e^{2y} \, dy.
     \]

4. **Evaluate the integral**:
   - The integral of \( e^{2y} \) is:
     \[
     \int e^{2y} \, dy = \frac{e^{2y}}{2}.
     \]
     Substituting the limits:
     \[
     V = 2\pi \left[ \frac{e^{2y}}{2} \right]_{0}^{1} = 2\pi \left( \frac{e^2}{2} - \frac{1}{2} \right).
     \]

5. **Simplify the result**:
   - Simplifying the expression:
     \[
     V = \pi \left( e^2 - 1 \right).
     \]

### Final Answer:
The volume of the solid \( S \) is:
\[
V = \pi \left( e^2 - 1 \right).
\]

Would you like any additional details or clarifications on this solution?

Here are 5 related questions to expand your understanding:
1. What is the method of cylindrical shells and when is it used in calculating volumes?
2. How would the solution change if the region was rotated around the x-axis instead of the y-axis?
3. What if the region was bounded by \( y = \ln(x) \) and a different value of \( y = c \)?
4. How do we derive the formula for the volume of a solid of revolution using the disk method?
5. Could you calculate the surface area of the same solid instead of the volume?

**Tip**: When calculating volumes of solids of revolution, make sure to choose the appropriate method (shells or disks) based on the axis of rotation and the shape of the region.

Let R be the region in the xy-plane bounded by the curves y = ln(x), y = 1, and x = 1. If S is the solid obtained by rotating R about the y-axis, find the volume of S.

Learn how to compute the volume of a solid obtained by rotating the region bounded by y = ln(x), y = 1, and x = 1 around the y-axis using the method of cylindrical shells. This problem demonstrates key calculus concepts related to volumes of solids of revolution.

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