To set up an integral for the volume of the solid generated by rotating the region bounded by $y = 4x - x^2$ and the X-axis about the line $x = 7$, we can use the cylindrical shell method. This method is appropriate when rotating around a vertical axis that is not the y-axis.

Step-by-step process:

1. Identify the boundaries:

   • The curve given is $y = 4x - x^2$, and it intersects the X-axis at $y = 0$.
   • Set $y = 4x - x^2 = 0$ to find the points of intersection: $x(4 - x) = 0 \implies x = 0 \quad \text{and} \quad x = 4.$
   • Thus, the region is bounded by $x = 0$ and $x = 4$ on the x-axis.

2. Formula for the cylindrical shell method: The volume of a cylindrical shell is given by: $V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx$ where:

   • Radius is the distance from $x$ to the axis of rotation $x = 7$, i.e., $|7 - x| = 7 - x$ (since $x \leq 7$).
   • Height is the value of the function, $y = 4x - x^2$.

3. Set up the integral: The integral for the volume is: $V = \int_{0}^{4} 2\pi (7 - x)(4x - x^2) \, dx.$

This integral represents the volume of the solid formed by rotating the region about the line $x = 7$.

Would you like a step-by-step calculation of this integral, or any clarifications?

Here are some follow-up questions to deepen your understanding:

1. What are the limits of integration for a volume of revolution problem using the disk/washer method instead of the shell method?
2. How does changing the axis of rotation from $x = 7$ to $x = 10$ affect the volume?
3. What if the curve was $y = 3x - x^2$ instead of $y = 4x - x^2$? How would the integral setup change?
4. Can this region be rotated about the y-axis using the disk method instead of shells? How would you set that up?
5. How do you interpret the meaning of "radius" in the context of cylindrical shells?

Tip: In the shell method, always think of the radius as the horizontal distance from the curve to the axis of rotation!