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To solve this problem, we need to find the volume of the solid when the region under the curve \( y = \frac{6}{x^2 + 3x + 2} \) from \( x = 0 \) to \( x = 1 \) is rotated about both the x-axis and the y-axis.

### 1. **Volume of Solid when Rotated about the x-axis:**
We can use the **disk method** for this case. The volume \( V_x \) of a solid of revolution generated by rotating a curve about the x-axis is given by:

\[
V_x = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]

Here, \( f(x) = \frac{6}{x^2 + 3x + 2} \), and the limits of integration are \( x = 0 \) to \( x = 1 \). Thus, we have:

\[
V_x = \pi \int_{0}^{1} \left(\frac{6}{x^2 + 3x + 2}\right)^2 dx
\]

### 2. **Volume of Solid when Rotated about the y-axis:**
To rotate the region about the y-axis, we use the **shell method**. The volume \( V_y \) of a solid generated by rotating a curve about the y-axis is given by:

\[
V_y = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]

Here, \( f(x) = \frac{6}{x^2 + 3x + 2} \) and the limits are again from \( x = 0 \) to \( x = 1 \). So, the volume is:

\[
V_y = 2\pi \int_{0}^{1} x \cdot \frac{6}{x^2 + 3x + 2} \, dx
\]

Now, I'll compute both integrals to find the volumes. Let's calculate them step by step.The volumes of the solids generated by rotating the region under the curve about the x-axis and the y-axis are as follows:

1. **Volume when rotated about the x-axis:**
   \[
   V_x = \pi \left( -144\ln(2) + 24 + 72\ln(3) \right)
   \]

2. **Volume when rotated about the y-axis:**
   \[
   V_y = 2\pi \left( -18\ln(2) + 12\ln(3) \right)
   \]

These are the exact forms of the volumes. If you'd like, I can approximate them numerically as well.

Would you like a numerical approximation or more details on the process? Here are five questions that expand upon this topic:

1. How does the shell method differ from the disk method in calculating volumes?
2. What role do logarithms play in the final form of these integrals?
3. How would the result change if the region was rotated about a different line (e.g., \( x = 2 \))?
4. How do you handle integrals involving rational functions like the one in this problem?
5. What happens to the volume if the bounds of integration are changed?

**Tip:** For volumes of revolution, always check whether the disk or shell method is easier based on the axis of rotation.

Find the volume of the resulting solid if the region under the curve y = 6 / (x^2 + 3x + 2) from x = 0 to x = 1 is rotated about the x-axis and the y-axis.

Learn how to calculate the volume of the solid formed by rotating the curve y = 6 / (x^2 + 3x + 2) from x = 0 to x = 1 about both the x-axis and y-axis. This problem involves the use of the disk and shell methods for finding the volume of solids of revolution, applicable in advanced calculus.

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