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

To find the volume of the solid generated when the region under the curve \( y = \frac{3}{x^2 + 3x + 2} \) is rotated about both the x-axis and the y-axis, we'll approach the problem in two parts:

### 1. Volume of Solid by Rotation about the x-axis (using the Disk Method)

The formula for the volume of a solid obtained by rotating a curve \( y = f(x) \) around the x-axis is:

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

For the given curve \( y = \frac{3}{x^2 + 3x + 2} \) and the limits \( x = 0 \) to \( x = 1 \), we first square the function and then integrate it:

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

### 2. Volume of Solid by Rotation about the y-axis (using the Shell Method)

The formula for the volume of a solid obtained by rotating a curve \( y = f(x) \) around the y-axis is:

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

Here, the limits are also \( x = 0 \) to \( x = 1 \), and \( f(x) = \frac{3}{x^2 + 3x + 2} \), so the integral becomes:

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

I'll now calculate both volumes using these formulas.

The volumes of the solids obtained by rotating the given curve about the x-axis and y-axis are as follows:

1. **Volume of the solid when rotated about the x-axis:**
   \[
   V_x = \pi \left( 6 - 36 \log(2) + 18 \log(3) \right)
   \]

2. **Volume of the solid when rotated about the y-axis:**
   \[
   V_y = 2\pi \left( -9 \log(2) + 6 \log(3) \right)
   \]

If you'd like further details or to simplify these expressions, feel free to ask!

Here are five follow-up questions you might explore:
1. How do we interpret the result if the curve is rotated around other axes?
2. What is the significance of the logarithmic terms in the volume formula?
3. Can the method be extended to non-polynomial functions?
4. How would the volume change if the limits of integration were extended?
5. How does the shell method compare with the disk method for other types of curves?

**Tip:** For problems involving volume of revolution, always check if the problem asks for rotation around a specific axis, as the method (disk or shell) depends on this choice.

Find the volume of the resulting solid if the region under the curve y = 3 / (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 the solid generated by rotating the curve y = 3 / (x^2 + 3x + 2) about the x-axis and y-axis. This calculus problem demonstrates the Disk and Shell Methods, involving integral calculus. Suitable for advanced high school or college-level students.

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