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We are asked to find the volume of the solid obtained by rotating the region under the curve:

\[
y = \frac{2}{x^2 + 3x + 2}
\]

over the interval \( x = 0 \) to \( x = 1 \) about both the x-axis and the y-axis.

### (a) Volume when rotated about the x-axis:
The formula for the volume of a solid of revolution about the x-axis is given by the disk method:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
where \( f(x) = y \), and the bounds are from \( x = 0 \) to \( x = 1 \).

Substitute the function \( y = \frac{2}{x^2 + 3x + 2} \) into the formula:

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

We'll compute this integral to find the volume.

### (b) Volume when rotated about the y-axis:
For rotation about the y-axis, we use the method of cylindrical shells, with the formula:

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

Substitute \( f(x) = \frac{2}{x^2 + 3x + 2} \):

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

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

### (a) Rotation about the x-axis:
\[
V = \pi \left( \frac{8}{3} - 16 \ln(2) + 8 \ln(3) \right)
\]

### (b) Rotation about the y-axis:
\[
V = 2\pi \left( 4 \ln(3) - 6 \ln(2) \right)
\]

If you need further details or clarifications, feel free to ask!

Here are five related questions:
1. What is the significance of the disk method in finding volumes of solids?
2. How does the method of cylindrical shells differ from the disk method?
3. Can you compute the volume for different intervals?
4. How would the volume change if the function were different, like \( y = \frac{1}{x+1} \)?
5. What happens if the rotation is done around a line other than the x- or y-axis?

**Tip:** When rotating around the x-axis, you're summing up disks (cross-sectional areas), while rotation around the y-axis involves summing cylindrical shells.

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

Calculate the volume of the solid formed by rotating the curve y = 2 / (x^2 + 3x + 2) from x = 0 to x = 1 around the x-axis and y-axis using integration techniques like the disk and shell methods.

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