Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. Ve results using the integration capabilities of a graphing utility. y = cos 3x y = 0 x = 0 X =- '이
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. Ve results using the integration capabilities of a graphing utility. y = cos 3x y = 0 x = 0 X =- '이
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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![### Calculating the Volume of a Solid of Revolution Using Integration
To find the volume of the solid generated by revolving the region bounded by the given equations around the x-axis, follow these steps. We will be using the integration capabilities of a graphing utility for verification.
#### Given Equations:
1. \( y = \cos 3x \)
2. \( y = 0 \)
3. \( x = 0 \)
4. \( x = \frac{\pi}{6} \)
#### Volume of Solid of Revolution:
The volume \( V \) of the solid formed by revolving a region around the x-axis can be determined using the Disk Method, which involves the integral:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Here, \( f(x) = \cos 3x \), and the limits of integration are from \( x = 0 \) to \( x = \frac{\pi}{6} \).
#### Set Up the Integral:
\[ V = \pi \int_{0}^{\frac{\pi}{6}} (\cos 3x)^2 \, dx \]
This integral can be evaluated using a graphing utility or integration techniques, such as trigonometric identities and substitution.
#### Verification:
Use the integration capabilities of your graphing utility to compute the integral and find the volume.
By setting up and solving this integral, you will find the volume of the solid formed by the given boundary conditions when revolved around the x-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad118f04-c8c0-46fe-8652-72d195094665%2Ff198b2ec-f20e-4da6-8a21-4b858071404c%2Fwm2c099_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Volume of a Solid of Revolution Using Integration
To find the volume of the solid generated by revolving the region bounded by the given equations around the x-axis, follow these steps. We will be using the integration capabilities of a graphing utility for verification.
#### Given Equations:
1. \( y = \cos 3x \)
2. \( y = 0 \)
3. \( x = 0 \)
4. \( x = \frac{\pi}{6} \)
#### Volume of Solid of Revolution:
The volume \( V \) of the solid formed by revolving a region around the x-axis can be determined using the Disk Method, which involves the integral:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Here, \( f(x) = \cos 3x \), and the limits of integration are from \( x = 0 \) to \( x = \frac{\pi}{6} \).
#### Set Up the Integral:
\[ V = \pi \int_{0}^{\frac{\pi}{6}} (\cos 3x)^2 \, dx \]
This integral can be evaluated using a graphing utility or integration techniques, such as trigonometric identities and substitution.
#### Verification:
Use the integration capabilities of your graphing utility to compute the integral and find the volume.
By setting up and solving this integral, you will find the volume of the solid formed by the given boundary conditions when revolved around the x-axis.
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