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.