Use the shell method to write and evaluate the de y = 8 - x -1 6- YA 2- x 4
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
Use shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis

Transcribed Image Text:**Transcription for Educational Website:**
Title: Using the Shell Method to Evaluate Volumes
Instructions: Use the shell method to write and evaluate the definite integral.
Equation: \( y = 8 - x \)
Response: \(-1\) ❌
---
**Diagram Explanation:**
The graph displays a triangular region under the line \( y = 8 - x \) on the coordinate plane. The triangle is shaded in blue and bounded by the x-axis from \( x = 0 \) to \( x = 8 \), and the y-axis.
- The hypotenuse of the triangle is the graph of the equation \( y = 8 - x \), which intersects the x-axis at \( (8, 0) \) and the y-axis at \( (0, 8) \).
- There is a vertical yellow rectangle within the triangle at \( x = 4 \), representing a typical shell or element used in setting up the integral for the shell method.
- The x-axis and y-axis are labeled, with the x-axis ranging from 0 to 8 and the y-axis from 0 to 8.
This setup is used to find the volume of a solid of revolution by rotating the triangular region around an axis, utilizing the shell method.
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