The exact volume of the solid of revolution in terms of \(\pi\) is:

\[
V = \pi \left(-164.811 + 288.477\sqrt{3} \right) \text{ cubic units}
\]

This expression gives the volume as a combination of a constant and a term involving \(\sqrt{3}\), both multiplied by \(\pi\).

Do you have any further questions or need more details?

Here are 8 additional related questions:

1. How can you factor the volume expression to better understand its components?
2. What is the approximate numerical value of \( 288.477\sqrt{3} \)?
3. How does expressing the volume in terms of \(\pi\) help in understanding the geometry of the solid?
4. How would the volume expression change if \( p(x) \) or \( q(x) \) were altered slightly?
5. Is there a scenario where the volume could be a simple multiple of \(\pi\)?
6. How can you verify the accuracy of the volume formula?
7. What is the significance of the negative term in the volume expression?
8. Can you derive a general formula for the volume of solids of revolution involving square roots in \( p(x) \) or \( q(x) \)?

**Tip:** When working with volumes in terms of \(\pi\), focus on the relationships between the coefficients to gain insights into the problem's structure.

Explore the exact volume of a solid of revolution expressed in terms of pi. This detailed explanation includes a formula combining constants and square roots multiplied by pi. Dive into related questions and insights.

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