Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Find Mx, My, and (x, y) for the laminas of uniform density ? bounded by the graphs of the equations.
![### Finding Moments and Centroid for Lamina of Uniform Density
Given the equations that bound the lamina with uniform density \(\rho\):
\[ y = \frac{1}{8}x \]
\[ y = 0 \]
\[ x = 8 \]
We are required to find the moments \(M_x\), \(M_y\), and the centroid \((\bar{x}, \bar{y})\).
#### Steps and Formulas
1. **Moment \(M_x\)**:
\[
M_x = \int_{a}^{b} y \, dA
\]
where \(dA\) is the differential element of area.
2. **Moment \(M_y\)**:
\[
M_y = \int_{a}^{b} x \, dA
\]
3. **Centroid \((\bar{x}, \bar{y})\)**:
\[
\bar{x} = \frac{M_y}{m}
\]
\[
\bar{y} = \frac{M_x}{m}
\]
where \(m\) is the total mass of the lamina.
#### Calculation Setup
\[
M_x = \boxed{\phantom{______}}
\]
\[
M_y = \boxed{\phantom{______}}
\]
\[
(\bar{x}, \bar{y}) = \left( \boxed{\phantom{______}}, \boxed{\phantom{______}} \right)
\]
The graphs and diagrams associated with the equations provided are not explicitly illustrated here but would typically include plotting each given equation to visualize the bounded region of the lamina. The boundaries are defined as follows:
- The straight line \(y = \frac{1}{8} x\),
- The x-axis \(y = 0\),
- The vertical line \(x = 8\).
This process involves setting up the integrals for \(M_x\) and \(M_y\) over the region defined by the given boundaries and then evaluating them to find the moments and subsequently the centroid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda6dd950-15a2-4e47-84e9-a8f156faff1f%2F5ec6f357-32d2-4093-acff-8add94e0137d%2Fdntuxes_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Moments and Centroid for Lamina of Uniform Density
Given the equations that bound the lamina with uniform density \(\rho\):
\[ y = \frac{1}{8}x \]
\[ y = 0 \]
\[ x = 8 \]
We are required to find the moments \(M_x\), \(M_y\), and the centroid \((\bar{x}, \bar{y})\).
#### Steps and Formulas
1. **Moment \(M_x\)**:
\[
M_x = \int_{a}^{b} y \, dA
\]
where \(dA\) is the differential element of area.
2. **Moment \(M_y\)**:
\[
M_y = \int_{a}^{b} x \, dA
\]
3. **Centroid \((\bar{x}, \bar{y})\)**:
\[
\bar{x} = \frac{M_y}{m}
\]
\[
\bar{y} = \frac{M_x}{m}
\]
where \(m\) is the total mass of the lamina.
#### Calculation Setup
\[
M_x = \boxed{\phantom{______}}
\]
\[
M_y = \boxed{\phantom{______}}
\]
\[
(\bar{x}, \bar{y}) = \left( \boxed{\phantom{______}}, \boxed{\phantom{______}} \right)
\]
The graphs and diagrams associated with the equations provided are not explicitly illustrated here but would typically include plotting each given equation to visualize the bounded region of the lamina. The boundaries are defined as follows:
- The straight line \(y = \frac{1}{8} x\),
- The x-axis \(y = 0\),
- The vertical line \(x = 8\).
This process involves setting up the integrals for \(M_x\) and \(M_y\) over the region defined by the given boundaries and then evaluating them to find the moments and subsequently the centroid.
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