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To solve the problem of finding the volume of the solid generated by rotating the region under the curve \( y = \frac{8}{x^2 + 3x + 2} \) around the x-axis and y-axis, we can apply two different methods:

### Part (a) – Rotation about the x-axis:
For rotation about the x-axis, we use the **disk method**, which gives the formula:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
where \( f(x) = \frac{8}{x^2 + 3x + 2} \), and we integrate from \( x = 0 \) to \( x = 1 \).

Thus, the volume is:
\[
V = \pi \int_{0}^{1} \left( \frac{8}{x^2 + 3x + 2} \right)^2 dx
\]

### Part (b) – Rotation about the y-axis:
For rotation about the y-axis, we use the **shell method**, which gives the formula:
\[
V = 2\pi \int_{a}^{b} x f(x) \, dx
\]
where \( f(x) = \frac{8}{x^2 + 3x + 2} \).

Thus, the volume is:
\[
V = 2\pi \int_{0}^{1} x \left( \frac{8}{x^2 + 3x + 2} \right) dx
\]

Let me calculate both integrals to determine the volumes.

Here are the results for the volumes:

### (a) Volume when the region is rotated around the x-axis:
\[
V_x = \pi \left( \frac{128}{3} - 256 \ln(2) + 128 \ln(3) \right)
\]

### (b) Volume when the region is rotated around the y-axis:
\[
V_y = 2\pi \left( 16 \ln(3) - 24 \ln(2) \right)
\]

These are the exact forms of the volumes, expressed with logarithms. If you'd like numerical approximations or further explanations, feel free to ask!

Would you like a detailed breakdown of the steps or any further clarifications?

Here are 5 related questions:
1. How does the disk method work for finding the volume of solids?
2. How does the shell method differ from the disk method?
3. What role does the natural logarithm play in the solution of these integrals?
4. How would the volume change if the region was rotated around a different axis?
5. Can this method be applied to other curves? If so, how?

**Tip:** Understanding the difference between the disk and shell methods is crucial for visualizing solids of revolution correctly!

Find the volume of the resulting solid if the region under the curve y = 8 / (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 a solid generated by rotating the curve y = 8 / (x^2 + 3x + 2) around the x-axis and y-axis. This involves applying the disk and shell methods, using key integration techniques from calculus. Suitable for university-level calculus courses.

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