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The problem asks to find the volume of the solid resulting from rotating the given curve \( y = \frac{5}{x^2 + 3x + 2} \) around the x-axis and y-axis, from \( x = 0 \) to \( x = 1 \).

### (a) Rotation around the x-axis:
For rotation around the x-axis, we use the disk method. The formula for the volume of a solid of revolution is:

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

Where \( f(x) \) is the given function \( y \).

So, for the given function:

\[
V = \pi \int_{0}^{1} \left( \frac{5}{x^2 + 3x + 2} \right)^2 \, dx
\]

### (b) Rotation around the y-axis:
For rotation around the y-axis, we use the shell method. The formula for the volume is:

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

So, for the given function:

\[
V = 2\pi \int_{0}^{1} x \cdot \frac{5}{x^2 + 3x + 2} \, dx
\]

Let me calculate these volumes step-by-step.The volumes of the solids formed by rotating the given region around the x-axis and y-axis are as follows:

### (a) Rotation around the x-axis:
\[
V = \pi \left( \frac{50}{3} - 100 \ln(2) + 50 \ln(3) \right)
\]

### (b) Rotation around the y-axis:
\[
V = 2\pi \left( -15 \ln(2) + 10 \ln(3) \right)
\]

These are the exact values in terms of natural logarithms. Would you like a numerical approximation for these results or any further clarifications?

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Here are 5 related questions:
1. How would the volume change if the limits of integration were different?
2. What happens if the curve is rotated around other axes, such as \( y = 1 \)?
3. Can we use the washer method if there is a gap between the curve and the axis?
4. How do you calculate surface area instead of volume for this type of problem?
5. What would be the interpretation of the result if we rotated the region around a slanted line?

**Tip:** For problems involving volumes of rotation, always verify if the disk, washer, or shell method is more appropriate depending on the axis of rotation.

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.

Learn how to calculate the volume of a solid of revolution formed by rotating the curve y = 5 / (x^2 + 3x + 2) from x = 0 to x = 1 about both the x-axis and the y-axis. This involves the disk and shell methods of integration, commonly taught in advanced high school calculus.

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