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 and y-axes

To find the volume of the solid formed by rotating the region under the curve \( y = \frac{3}{x^2 + 3x + 2} \) from \( x = 0 \) to \( x = 1 \) about both the **x-axis** and the **y-axis**, we'll use the methods of disk/washer and shell integration, respectively.

### 1. Volume of the solid rotated about the x-axis (Disk/Washer method)

Using the disk/washer method, the formula for the volume of revolution about the x-axis is:

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

For the given function, \( f(x) = \frac{3}{x^2 + 3x + 2} \), and the limits of integration are \( x = 0 \) to \( x = 1 \).

Thus, the volume \( V_x \) is:

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

This integral is somewhat complex, so I will evaluate it for you.

### 2. Volume of the solid rotated about the y-axis (Shell method)

For rotation about the y-axis, we use the cylindrical shell method. The volume formula is:

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

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

Thus, the volume \( V_y \) is:

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

I'll evaluate both integrals to find the respective volumes.

Let's compute these integrals.The volumes of the solids are as follows:

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

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

These expressions involve logarithmic terms and represent the exact volumes.

Would you like to see these volumes as decimal approximations, or do you have any other questions?

### Related questions:
1. How does the method of cylindrical shells work for more general curves?
2. Can this problem be solved if rotated about different axes (e.g., lines like \( x = 2 \))?
3. What if the curve had a different function in the denominator?
4. How does the disk method compare to the washer method in terms of applications?
5. What happens to the volumes if the region is rotated around an arbitrary line parallel to the x-axis or y-axis?

**Tip**: For complex integrals, logarithmic terms often arise, especially when dealing with rational functions like this one.

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 and y-axes.

Learn how to find the volume of a solid formed by rotating the region under the curve y = 3/(x^2 + 3x + 2) from x = 0 to x = 1 around the x-axis and y-axis. This problem covers integration techniques using the disk/washer and shell methods in calculus, suitable for college-level students.

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