To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves $x = y^2$, $y = 0$, and $x = 4$ about the line $x = 9$, we will use the shell method because the axis of rotation, $x = 9$, is parallel to the region but not on the boundary.

Step-by-step setup using the shell method:

1. Visualize the region:

   • The curve $x = y^2$ is a parabola opening to the right.
   • The region is bounded by the line $x = 4$ (a vertical line) and the horizontal line $y = 0$.
   • This region will be rotated about the vertical line $x = 9$.

2. Shell radius: The distance from the line of rotation $x = 9$ to a point on the curve $x = y^2$ is: $\text{Radius of shell} = 9 - x = 9 - y^2$ where $x = y^2$.

3. Shell height: The height of each shell is the horizontal distance between the line $x = y^2$ and the vertical boundary at $x = 4$. So, the height is: $\text{Height of shell} = 4 - y^2$

4. Volume element (dV): The volume element using the shell method is: $dV = 2\pi \times (\text{radius of shell}) \times (\text{height of shell}) \times dy$ Substituting in the expressions for the radius and height: $dV = 2\pi (9 - y^2)(4 - y^2) \, dy$

5. Limits of integration: The limits for $y$ come from the boundaries of the region. From the curve $x = y^2$, we know: $y = 0 \quad \text{to} \quad y = 2$ (since at $x = 4$, $y = \sqrt{4} = 2$).

6. Integral setup: The integral for the volume is: $V = \int_{0}^{2} 2\pi (9 - y^2)(4 - y^2) \, dy$

This integral represents the volume of the solid obtained by rotating the region about $x = 9$.

Would you like more details or have any questions about the setup?

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Here are 5 related questions that build on this topic:

1. How would the integral change if the rotation axis were $x = 0$?
2. Can you use the disk/washer method for this problem instead? If so, how would you set it up?
3. What if the boundary was extended to $x = 9$? How would that affect the integral?
4. How does changing the rotation axis to a different line, like $x = 5$, affect the volume?
5. How would the setup change if the region were rotated around the y-axis instead?

Tip: The shell method is especially useful when rotating around a vertical line that is not the boundary of the region, as in this case!