https://file.aiquickdraw.com/mathbot/file/1726628286052ay6n2stm.jpg

The problem in the image asks to find the volume of the resulting solid when the region under the curve 

\[
y = \frac{5}{x^2 + 3x + 2}
\]

from \( x = 0 \) to \( x = 1 \) is rotated about:

1. The **x-axis** (part a).
2. The **y-axis** (part b, not shown but implied).

### Part (a): Volume when rotated about the **x-axis**

To find the volume of the solid when the curve is rotated about the **x-axis**, we use the formula for the volume of revolution about the x-axis:

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

Here:
- \( f(x) = \frac{5}{x^2 + 3x + 2} \)
- The limits of integration are from \( x = 0 \) to \( x = 1 \).

Thus, the volume is:

\[
V = \pi \int_{0}^{1} \left( \frac{5}{x^2 + 3x + 2} \right)^2 dx
\]

We can now solve this integral to find the volume. Let's compute it.The volume of the solid when the region under the curve \( y = \frac{5}{x^2 + 3x + 2} \) from \( x = 0 \) to \( x = 1 \) is rotated about the **x-axis** is:

\[
V = \frac{50\pi}{3} \left( \log\left(\frac{27}{64}\right) + 1 \right)
\]

Would you like further details or to explore the case where the solid is rotated about the y-axis?

Here are five related questions for deeper understanding:
1. What method is used for finding the volume when rotating about the y-axis?
2. How do we handle integration when dealing with more complex functions?
3. Can we use numerical methods to approximate the volume if the integral is hard to compute?
4. What is the significance of using the disk/washer method for solid volumes?
5. How does the region bounded by different functions affect the volume?

**Tip:** Always double-check limits of integration when setting up problems involving areas or volumes of rotation.

Find the volume of the resulting solid if the region under the curve y = 5 / (x^2 + 3x + 2) from x = 0 to x = 1 is rotated about the x-axis.

Learn how to compute the volume of a solid when the curve y = 5 / (x^2 + 3x + 2) is rotated about the x-axis between x = 0 and x = 1. This problem demonstrates the application of the disk method and requires integration of a rational function.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation