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To solve the given problem, we need to find the volume of the resulting solid when the region under the curve 

\[
y = \frac{5}{x^2 + 3x + 2}
\]

from \(x = 0\) to \(x = 1\) is rotated about both the x-axis and the y-axis.

### (a) Rotation about the **x-axis**:
We use the **Disk Method** to find the volume of the solid obtained by rotating the region about the x-axis. The formula for the volume of a solid of revolution around the x-axis is:

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

Here, \( f(x) = \frac{5}{x^2 + 3x + 2} \), and we integrate from \(x = 0\) to \(x = 1\).

So the volume is:

\[
V = \pi \int_0^1 \left( \frac{5}{x^2 + 3x + 2} \right)^2 \, dx
\]

### (b) Rotation about the **y-axis**:
For rotation about the y-axis, we use the **Shell Method**. The formula for the volume of a solid of revolution around the y-axis is:

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

So the volume is:

\[
V = 2\pi \int_0^1 x \cdot \frac{5}{x^2 + 3x + 2} \, dx
\]

Let me compute both integrals for you.Here are the results for the volumes:

### (a) Volume when rotated about the x-axis:
\[
V_x = 25\pi \left( -2\ln(2) - \frac{5}{6} + 2\ln(3) \right) - 25\pi \left( -\frac{3}{2} + 2\ln(2) \right)
\]

### (b) Volume when rotated about the y-axis:
\[
V_y = -20\pi\ln(2) + 10\pi \left( -\ln(2) + 2\ln(3) \right)
\]

Would you like a numerical approximation for these volumes or further details on how these expressions were derived? Let me know if you have any questions!

Here are 5 related questions for further exploration:
1. How would the volume change if the limits of integration were different?
2. What happens to the volume if the curve is rotated about the line \(y = c\)?
3. Can the method of cylindrical shells be applied to any other axis?
4. How do you calculate the surface area of the solid of revolution?
5. What if we rotate the region about a vertical line other than the y-axis?

**Tip**: Always double-check the limits of integration when applying rotation methods for solids to avoid mistakes in calculations.

Find the volume of the resulting solid if the region under the curve y = 5 / (x^2 + 3x + 2) from x = 0 to x = 1 is rotated about the x-axis and the y-axis.

Find the volume of the solid formed by rotating the curve y = 5 / (x^2 + 3x + 2) from x = 0 to x = 1 about the x-axis and y-axis. This problem involves calculus techniques such as the Disk Method and Shell Method for calculating the volume of solids of revolution.

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