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The image contains a problem asking to find the volume of the solid formed by rotating the region under the curve \( y = \frac{7}{x^2 + 3x + 2} \) from \( x = 0 \) to \( x = 1 \) about two axes:

1. Part (a) asks for the volume when rotating around the **x-axis**.
2. Part (b) asks for the volume when rotating around the **y-axis**.

### Solution Approach:

#### (a) Rotating Around the **x-axis**:

When rotating around the **x-axis**, the volume of the solid can be found using the **disk method**:

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

In this case, the curve is \( y = \frac{7}{x^2 + 3x + 2} \), so we will square \( y \) and integrate from \( x = 0 \) to \( x = 1 \):

\[
V = \pi \int_0^1 \left( \frac{7}{x^2 + 3x + 2} \right)^2 dx
\]

However, the solution provided seems to show a complicated logarithmic answer involving a natural logarithm and terms with \(\ln\left(\frac{3}{4}\right)\).

To compute this integral exactly would require using partial fractions or a computer algebra system. The provided solution in part (a) seems to be correct, as indicated by the cross mark being near part (b).

#### (b) Rotating Around the **y-axis**:

This part likely involves the **shell method** because the volume of revolution around the **y-axis** requires integrating with respect to \( x \) while incorporating the distance from the axis of rotation:

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

Using \( f(x) = \frac{7}{x^2 + 3x + 2} \), we would set up the integral as:

\[
V = 2\pi \int_0^1 x \cdot \frac{7}{x^2 + 3x + 2} \, dx
\]

This integral can also be solved using partial fractions or another appropriate technique.

Would you like me to go into more detail about the integration process or the specific methods used for these problems?

### 5 Related Questions:

1. How do you apply the shell method to other functions when rotating around the y-axis?
2. What are the differences between the disk and shell methods for finding volumes?
3. How do partial fractions help in solving integrals involving rational functions?
4. What are common techniques for evaluating integrals with logarithms?
5. How do you visualize the solid formed by rotating a region around the x-axis or y-axis?

### Tip:
When rotating a region around an axis, always carefully decide whether to use the disk/washer method (for horizontal axes) or the shell method (for vertical axes).

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.

This problem involves finding the volume of a solid formed by rotating the region under the curve y = 7 / (x^2 + 3x + 2) from x = 0 to x = 1 around both the x-axis and y-axis. Using the disk and shell methods, learn how to compute the volumes step-by-step.

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