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To solve this problem, we will use the method of cylindrical shells. We are rotating the region in the first quadrant bounded by the curves \( x = 0 \), \( y = 1 \), \( x = y^3 \), about the line \( y = 1 \).

### Step-by-step solution:

1. **Understanding the region**:
   - The curve \( x = y^3 \) and the line \( y = 1 \) bound the region, with \( x = 0 \) as the left boundary. We are rotating around the horizontal line \( y = 1 \).
   
2. **Set up the volume integral using cylindrical shells**:
   - For a solid generated by rotating about a horizontal line, the height of each shell is based on the \( x \)-coordinate of the boundary, and the radius of the shell is the vertical distance from the line of rotation.

   - The radius of the shell is \( r(y) = 1 - y \) (since the rotation is about \( y = 1 \)).
   
   - The height of the shell is determined by the boundary curve \( x = y^3 \).
   
   - The volume of a cylindrical shell is given by:
     \[
     dV = 2\pi \cdot (\text{radius}) \cdot (\text{height}) \cdot (\text{thickness})
     \]
     So, the volume element is:
     \[
     dV = 2\pi \cdot (1 - y) \cdot y^3 \, dy
     \]

3. **Determine limits of integration**:
   - The region is bounded by \( y = 0 \) and \( y = 1 \).

4. **Set up and compute the integral**:
   The total volume \( V \) is:
   \[
   V = \int_0^1 2\pi (1 - y) y^3 \, dy
   \]
   We can now solve this integral step by step.

Let's proceed to compute the integral.The volume of the solid obtained by rotating the region around the line \( y = 1 \) is \( \frac{\pi}{10} \).

Would you like further details or have any other questions?

Here are five related questions for deeper understanding:
1. How would the solution change if the region were rotated around the \( x \)-axis?
2. What happens if we rotate the region around a different horizontal line, such as \( y = 2 \)?
3. Can this problem be solved using the disk/washer method instead of cylindrical shells?
4. What is the volume if the curve \( x = y^3 \) is replaced with another function, such as \( x = y^2 \)?
5. How would the limits of integration change if the region extended into negative \( y \)-values?

**Tip:** The method of cylindrical shells is often useful when the axis of rotation is parallel to the boundaries of the region you're integrating.

Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves x=0, y=1, x=y^3, about the line y=1.

Learn how to calculate the volume of a solid obtained by rotating the region bounded by x=0, y=1, and x=y^3 about the line y=1 using the cylindrical shells method. This problem covers advanced calculus topics suitable for high school students in grades 11-12.

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