Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y=x+12 and the parabola y=x^2 about the line x=4. Don't show any steps, just give me the final answer. Keep it in terms of pi.

The volume of the solid is \( 320\pi \) cubic units. 

Do you want details or have any questions? Here are 5 related questions:

1. How would the volume change if the region was revolved about the y-axis instead?
2. Can you use the disk/washer method to find the same volume?
3. What is the significance of the shell method compared to the disk method?
4. How would the bounds of integration change if the line was \(x = 3\) instead of \(x = 4\)?
5. What happens to the volume if the parabola is shifted upwards?

**Tip:** The shell method is often more convenient when the axis of rotation is parallel to the axis that defines the height of the region.

Find the volume of the solid formed by revolving the region bounded by the line y=x+12 and the parabola y=x^2 about the line x=4 using the shell method. The final answer is in terms of pi. Suitable for advanced calculus students in Grades 11-12.

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