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The problem asks to find the volume of the resulting solid when the region under the curve:

\[
y = \frac{7}{x^2 + 3x + 2}
\]

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

### 1. Volume of Solid by Rotation Around the X-Axis (Disk Method):

The formula for the volume of a solid of revolution about the x-axis is:

\[
V_x = \pi \int_{a}^{b} \left[ f(x) \right]^2 dx
\]

Substituting \(y = \frac{7}{x^2 + 3x + 2}\), we need to compute:

\[
V_x = \pi \int_{0}^{1} \left( \frac{7}{x^2 + 3x + 2} \right)^2 dx
\]

### 2. Volume of Solid by Rotation Around the Y-Axis (Shell Method):

The formula for the volume of a solid of revolution about the y-axis is:

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

Substituting \(y = \frac{7}{x^2 + 3x + 2}\), we need to compute:

\[
V_y = 2\pi \int_{0}^{1} x \cdot \frac{7}{x^2 + 3x + 2} \, dx
\]

Let me calculate both integrals for you.The volumes of the solid obtained by rotating the given region around the x-axis and the y-axis are:

1. **Volume around the x-axis (Disk method):**
   \[
   V_x = \pi \left( \frac{98}{3} - 196 \log(2) + 98 \log(3) \right)
   \]

2. **Volume around the y-axis (Shell method):**
   \[
   V_y = 2\pi \left( -21 \log(2) + 14 \log(3) \right)
   \]

Let me know if you'd like further details or additional steps explained!

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Here are 5 related questions you might want to explore:

1. How is the disk method derived for finding the volume of a solid?
2. What is the difference between the disk and shell methods in general?
3. How do logarithmic terms arise in integrals of rational functions?
4. What conditions affect whether an integral of a function has a closed-form solution?
5. How does rotation around different axes affect the shape of the solid formed?

**Tip:** The shell method is often preferable when the function is easier to integrate in terms of \(x\), but the distance from the axis of rotation (usually \(y\)) varies more directly with \(x\).

Find the volume of the resulting solid if the region under the curve y = 7 / (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 generated when the region under the curve y = 7 / (x^2 + 3x + 2) is rotated about the x-axis and the y-axis, using the disk and shell methods. This problem covers key calculus concepts, including integration and volumes of solids of revolution.

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