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The problem asks to set up (but not evaluate) an integral for the volume of the solid obtained by rotating the region bounded by the curves \( y = x^3 \), \( y = 0 \), \( x = 1 \), and rotating about the line \( x = 2 \).

### Step-by-step solution:

1. **Identify the method to use**: 
   - Since we are rotating about a vertical line \( x = 2 \), we use the **shell method**.
   
   The shell method formula is:
   \[
   V = 2\pi \int_{a}^{b} ( \text{radius} )(\text{height}) \, dx
   \]
   
2. **Radius of the shell**: 
   - The radius is the distance from the curve \( x = 2 \) to the current \( x \)-coordinate.
   - This distance is \( 2 - x \).

3. **Height of the shell**: 
   - The height is the value of the function \( y = x^3 \), which is just \( x^3 \).

4. **Bounds of integration**: 
   - The region is bounded by \( x = 0 \) and \( x = 1 \), so these will be the limits of integration.

5. **Setting up the integral**:
   \[
   V = 2\pi \int_{0}^{1} (2 - x)(x^3) \, dx
   \]

This is the required integral to compute the volume of the solid obtained by rotating the region around the line \( x = 2 \).

Would you like further details or have any questions?

### Related Questions:
1. What is the shell method, and when is it used instead of the disk/washer method?
2. How would the setup change if we were rotating about the \( y \)-axis instead of \( x = 2 \)?
3. How would we compute the volume if we used the disk/washer method (if applicable)?
4. What would be the integral setup if the function were \( y = x^2 \) instead of \( y = x^3 \)?
5. How does the choice of the axis of rotation affect the integral setup?

### Tip:
Always double-check the axis of rotation and whether the shell or disk method is more appropriate based on the problem's geometry.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the curves y = x^3, y = 0, x = 1, about x = 2.

Learn how to set up an integral using the shell method to find the volume of a solid obtained by rotating the region bounded by y = x^3, y = 0, x = 1 around the line x = 2. This problem covers key calculus concepts including the shell method and solids of revolution.

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