https://file.aiquickdraw.com/mathbot/file/1726619089231f9a2ohk1.jpg

To solve this problem, we are asked to find the volume of a solid of revolution for the given function \( y = \frac{3}{x^2 + 3x + 2} \), rotated about both the x-axis and the y-axis between \( x = 0 \) and \( x = 1 \).

### (a) Volume when rotated about the **x-axis**:
The volume of the solid formed by rotating the region around the x-axis can be found using the **disk method**. The formula for the volume is:

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

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

The volume is:

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

I'll calculate this integral to find the volume.

### (b) Volume when rotated about the **y-axis**:
To rotate the region around the y-axis, we use the **shell method**. The formula for the volume using cylindrical shells is:

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

In this case, the volume is given by:

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

Let me calculate these integrals now.The volumes of the solid obtained by rotating the region under the curve \( y = \frac{3}{x^2 + 3x + 2} \) are:

### (a) Around the **x-axis**:
\[
V_x = \pi \left( 6 - 36 \ln(2) + 18 \ln(3) \right)
\]

### (b) Around the **y-axis**:
\[
V_y = 2\pi \left( -9 \ln(2) + 6 \ln(3) \right)
\]

Would you like further details or assistance on these calculations?

Here are 5 related questions to expand your understanding:
1. How do you set up integrals for solids of revolution around different axes?
2. What is the difference between the disk and shell methods for volume?
3. How do logarithmic terms appear in volume integrals for solids of revolution?
4. Can you apply the same methods for more complex functions?
5. What are the conditions for switching between the disk and shell methods?

**Tip:** When rotating around the y-axis, the shell method is usually more convenient since it accounts for the horizontal radius, whereas the disk method is typically simpler for the x-axis rotation.

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.

Calculate 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 about the x-axis and y-axis. Learn how to apply the disk and shell methods using definite integrals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation