Find the volume of the solid generated when the region under the curve y = 1/(x2 + 3x + 2) from x = 0 to x = 1 is revolved about the given axis. Make certain to include sketches and justifications for each solid. 3. x-axis 4. y-axis
Find the volume of the solid generated when the region under the curve y = 1/(x2 + 3x + 2) from x = 0 to x = 1 is revolved about the given axis. Make certain to include sketches and justifications for each solid. 3. x-axis 4. y-axis
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Part 2**
Find the volume of the solid generated when the region under the curve \( y = \frac{1}{(x^2 + 3x + 2)} \) from \( x = 0 \) to \( x = 1 \) is revolved about the given axis. Make certain to include sketches and justifications for each solid.
3. x-axis
4. y-axis
**Explanation (for reference, not visible in image):**
For the x-axis:
- Use the disk or washer method. The volume can be calculated using the integral:
\[
V = \pi \int_{0}^{1} \left(\frac{1}{x^2 + 3x + 2}\right)^2 \, dx
\]
For the y-axis:
- Use the shell method. The volume is given by the integral:
\[
V = 2\pi \int_{0}^{1} x \left(\frac{1}{x^2 + 3x + 2}\right) \, dx
\]
In both cases, ensure that sketches show the curve, the axis of revolution, and the resulting solid shape.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc466199-e56c-4dcd-8ebf-897fec67301e%2F527fa8ff-2467-445a-924f-73d133f69ce7%2Fhlmn71_processed.png&w=3840&q=75)
Transcribed Image Text:**Part 2**
Find the volume of the solid generated when the region under the curve \( y = \frac{1}{(x^2 + 3x + 2)} \) from \( x = 0 \) to \( x = 1 \) is revolved about the given axis. Make certain to include sketches and justifications for each solid.
3. x-axis
4. y-axis
**Explanation (for reference, not visible in image):**
For the x-axis:
- Use the disk or washer method. The volume can be calculated using the integral:
\[
V = \pi \int_{0}^{1} \left(\frac{1}{x^2 + 3x + 2}\right)^2 \, dx
\]
For the y-axis:
- Use the shell method. The volume is given by the integral:
\[
V = 2\pi \int_{0}^{1} x \left(\frac{1}{x^2 + 3x + 2}\right) \, dx
\]
In both cases, ensure that sketches show the curve, the axis of revolution, and the resulting solid shape.
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