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To solve this problem, let's go through the steps to find the volume of the solid of revolution formed by revolving the region \( R \) around the \( y \)-axis. The region \( R \) is bounded by \( f(x) = \frac{1}{x} \), the \( x \)-axis, and the interval \( [1,3] \).

### Solution Outline:

1. **Identify the function and limits**: 
   - We are given \( f(x) = \frac{1}{x} \) from \( x = 1 \) to \( x = 3 \).
   
2. **Volume of Revolution around the \( y \)-axis**:
   - The formula for the volume of a solid of revolution about the \( y \)-axis is:
     \[
     V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
     \]
   - Here, \( f(x) = \frac{1}{x} \), so the volume integral becomes:
     \[
     V = 2\pi \int_{1}^{3} x \cdot \frac{1}{x} \, dx
     \]

3. **Simplify the Integral**:
   - Simplifying \( x \cdot \frac{1}{x} \) gives \( 1 \), so we have:
     \[
     V = 2\pi \int_{1}^{3} 1 \, dx
     \]
   - This integral now becomes:
     \[
     V = 2\pi \left[ x \right]_{1}^{3}
     \]

4. **Evaluate the Integral**:
   - Substitute the limits:
     \[
     V = 2\pi (3 - 1) = 2\pi \cdot 2 = 4\pi
     \]

### Answer:
The volume of the solid of revolution is \( \boxed{4\pi} \), which corresponds to option (a).

Would you like further details on any of these steps, or do you have other questions?

Here are some follow-up questions to consider:
1. How would the solution change if we were revolving around the \( x \)-axis instead?
2. What is the significance of the function \( f(x) = \frac{1}{x} \) in calculus problems involving areas and volumes?
3. How would this calculation differ if we used the shell method around a different axis?
4. Could we compute the same volume using a different method, like the disk/washer method?
5. How does changing the interval affect the resulting volume of revolution?

**Tip**: When dealing with solids of revolution, carefully choose between the disk/washer method and the shell method, depending on the axis of rotation and the function's form.

Define R as the region bounded above by the graph of f(x) = 1/x and below by the x-axis over the interval [1,3]. Find the volume of the solid of revolution formed by revolving R around the y-axis.

This calculus problem explores how to calculate the volume of a solid of revolution formed by rotating the region bounded by f(x) = 1/x, the x-axis, and the interval [1, 3] around the y-axis using the shell method.

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