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The problem asks to find the volume of the solid obtained by rotating the region under the curve \( y = \frac{4}{x^2 + 3x + 2} \) from \( x = 0 \) to \( x = 1 \) around both the x-axis and the y-axis. Let's solve both parts separately:

### 1. **Rotation about the x-axis:**

For rotation about the x-axis, we use the disk method. The volume \( V \) is given by the formula:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
where \( f(x) \) is the function describing the curve, and \( a \) and \( b \) are the limits of integration.

Here, \( f(x) = \frac{4}{x^2 + 3x + 2} \), and the limits are from \( x = 0 \) to \( x = 1 \).

Thus, the volume is:
\[
V = \pi \int_{0}^{1} \left( \frac{4}{x^2 + 3x + 2} \right)^2 \, dx
\]

### 2. **Rotation about the y-axis:**

For rotation about the y-axis, we use the shell method. The volume \( V \) is given by the formula:
\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]
where \( f(x) = \frac{4}{x^2 + 3x + 2} \), and the limits are the same, from \( x = 0 \) to \( x = 1 \).

Thus, the volume is:
\[
V = 2\pi \int_{0}^{1} x \cdot \frac{4}{x^2 + 3x + 2} \, dx
\]

Let me now calculate both integrals.The volumes for the two rotations are as follows:

1. **Volume when rotated about the x-axis (disk method):**
   \[
   V_x = \pi \left( \frac{32}{3} - 64 \ln(2) + 32 \ln(3) \right)
   \]

2. **Volume when rotated about the y-axis (shell method):**
   \[
   V_y = 2\pi \left( -12 \ln(2) + 8 \ln(3) \right)
   \]

If you'd like, I can provide more detailed steps or help with further questions.

Here are five questions for further exploration:
1. What is the geometric interpretation of rotating a curve around an axis?
2. How would the result change if the limits of integration were different?
3. What happens to the volume if the function is rotated about another line, such as \( x = 2 \)?
4. Can you visualize the shape created by rotating the curve around the axes?
5. How do you handle improper integrals when calculating volumes of solids?

**Tip:** When solving problems involving rotations, always carefully choose the appropriate method—disk/washer or shell—based on the axis of rotation.

Find the volume of the resulting solid if the region under the curve y = 4 / (x^2 + 3x + 2) from x = 0 to x = 1 is rotated about the x-axis and the y-axis.

This problem calculates the volume of a solid formed by rotating the region under the curve y = 4 / (x^2 + 3x + 2) from x = 0 to x = 1 around both the x-axis and y-axis. The solution uses the disk and shell methods to find the volume.

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