Consider the region bounded by y=x^3, y=0, and x=1. Find the volume of the solid obtained by rotating the region around the vertical line x=2

We are tasked with finding the volume of the solid formed by rotating the region bounded by \( y = x^3 \), \( y = 0 \), and \( x = 1 \) around the vertical line \( x = 2 \). To solve this, we will use the **method of cylindrical shells**.

### Step-by-Step Solution:

1. **General Setup for Cylindrical Shells**:
   When rotating a region around a vertical line not passing through the region itself, the volume can be found using the formula for the volume of cylindrical shells:

   \[
   V = 2\pi \int_{a}^{b} \left( \text{radius} \right) \cdot \left( \text{height} \right) \, dx
   \]
   
   Here:
   - The **radius** of a typical shell is the distance from \( x \) to the line \( x = 2 \), which is \( 2 - x \).
   - The **height** of a typical shell is given by the function \( y = x^3 \).
   
2. **Define the Limits of Integration**:
   The region is bounded by \( x = 0 \) and \( x = 1 \), so the limits of integration are from \( 0 \) to \( 1 \).

3. **Set Up the Integral**:
   Using the formula for the volume of cylindrical shells, we have:

   \[
   V = 2\pi \int_{0}^{1} (2 - x)(x^3) \, dx
   \]

4. **Simplify the Integrand**:
   Expand the expression inside the integral:

   \[
   V = 2\pi \int_{0}^{1} \left( 2x^3 - x^4 \right) \, dx
   \]

5. **Integrate**:
   Now integrate term by term:

   \[
   V = 2\pi \left[ \frac{2x^4}{4} - \frac{x^5}{5} \right]_{0}^{1}
   \]

   Simplify the expressions inside the brackets:

   \[
   V = 2\pi \left[ \frac{1}{2} - \frac{1}{5} \right]
   \]

6. **Compute the Final Value**:
   Simplify the expression:

   \[
   V = 2\pi \left( \frac{5}{10} - \frac{2}{10} \right) = 2\pi \cdot \frac{3}{10} = \frac{6\pi}{10} = \frac{3\pi}{5}
   \]

Thus, the volume of the solid is:

\[
\boxed{\frac{3\pi}{5}}
\]

Would you like further clarification or a more detailed breakdown of any of the steps?

### Related Questions:
1. How would the solution change if the rotation was around the line \( x = 3 \)?
2. Can we use the disk/washer method instead of cylindrical shells for this problem?
3. What if the region were rotated around the horizontal line \( y = -1 \)?
4. How do we approach this problem if the region is instead bounded by \( y = x^2 \)?
5. How do we apply the cylindrical shell method when rotating around \( x = -1 \)?

**Tip**: When using the method of cylindrical shells, the radius is always the distance from the rotation axis to the variable \( x \), which must be correctly defined depending on the position of the axis.

Consider the region bounded by y=x^3, y=0, and x=1. Find the volume of the solid obtained by rotating the region around the vertical line x=2.

Find the volume of the solid obtained by rotating the region bounded by y=x^3, y=0, and x=1 around the vertical line x=2. This solution applies the cylindrical shell method with detailed steps, including setting up and solving the integral. Suitable for students in Grades 11-12.

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