https://file.aiquickdraw.com/mathbot/file/1726611201811lh1qv53j.jpg

Let's solve the problem in two parts, as the question asks for the volume of the solid when the given region is rotated about both the **x-axis** and **y-axis**.

### Given Function:
\[
y = \frac{6}{x^2 + 3x + 2}
\]
From \(x = 0\) to \(x = 1\).

#### (a) Rotation around the **x-axis**:
We will use the **disk method** for this. The formula for the volume \(V\) of a solid of revolution about the x-axis is:

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

In this case, \(f(x) = \frac{6}{x^2 + 3x + 2}\), and the limits are from 0 to 1.

So the volume is:

\[
V = \pi \int_0^1 \left( \frac{6}{x^2 + 3x + 2} \right)^2 dx
\]

Let's calculate this integral step by step.

#### (b) Rotation around the **y-axis**:
This part has already been computed in the image, and the answer given is:

\[
V = 2\pi \left( -18\ln(2) + 12\ln(3) \right)
\]

This is correct based on the shell method formula.

---

I will calculate the integral for part (a) now.The volume of the solid when the region is rotated around the **x-axis** is:

\[
V = 24\pi \left( \ln\left(\frac{27}{64}\right) + 1 \right)
\]

Would you like further clarification on this solution, or help with any other aspect?

Here are some related questions for further understanding:

1. How is the disk method applied to solids of revolution about the x-axis?
2. How does the shell method differ from the disk method when revolving around the y-axis?
3. What is the significance of logarithms in this type of integral?
4. How would the integral change if the limits were different?
5. Could this method be applied to other curves with different forms?

**Tip**: The choice between disk and shell method depends on the axis of rotation and how the function is expressed.

Find the volume of the resulting solid if the region under the curve y = 6 / (x^2 + 3x + 2) from x = 0 to x = 1 is rotated about the x-axis and the y-axis.

Calculate the volume of a solid generated by rotating the curve y = 6 / (x^2 + 3x + 2) from x = 0 to x = 1 about both the x-axis and y-axis using disk and shell methods. This problem involves calculus concepts including integration and solid of revolution techniques.

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