Represent the moments of the region R bounded by the function f(x) = 1 + sin(x) and the x-axis and the lines x = 1 and x = 4. Mass = M = Moment about y-axis = My = = 4 X = S Moment about x-axis = Mx= = y = 4 S 1 Then calculate the center of mass for R (round to 2 decimal places) న 4 Sª dx dx dx
Represent the moments of the region R bounded by the function f(x) = 1 + sin(x) and the x-axis and the lines x = 1 and x = 4. Mass = M = Moment about y-axis = My = = 4 X = S Moment about x-axis = Mx= = y = 4 S 1 Then calculate the center of mass for R (round to 2 decimal places) న 4 Sª dx dx dx
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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![### Moments of the Region R
Consider the region R bounded by the function \( f(x) = 1 + \sin(x) \) and the x-axis, and the lines \( x = 1 \) and \( x = 4 \).
1. **Mass of the Region R:**
\[
\text{Mass} = M = \int_{1}^{4} (1 + \sin(x)) \, dx
\]
2. **Moment about y-axis:**
\[
M_y = \int_{1}^{4} x(1 + \sin(x)) \, dx
\]
3. **Moment about x-axis:**
\[
M_x = \int_{1}^{4} \frac{1 + \sin(x)}{2}(1 + \sin(x)) \, dx
\]
4. **Center of Mass:**
To find the center of mass, you need to calculate:
\[
\bar{x} = \frac{M_y}{M}
\]
\[
\bar{y} = \frac{M_x}{M}
\]
Make sure to round your final answers to two decimal places.
### Steps to Follow
- **Sketch the Region:**
Sketch the region defined by \( f(x) = 1 + \sin(x) \), the x-axis, and the vertical lines \( x = 1 \) and \( x = 4 \). Label the strip used for integration and the center of the strip.
- **Write the Definite Integrals:**
Set up the definite integrals for mass, and the moments about the x-axis and y-axis.
- **Evaluate the Integrals:**
Solve the definite integrals to find the moments and hence the center of mass (centroid) of the region R.
Use these steps to ensure a detailed understanding and accurate calculation of the moments and centroid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2386874c-e724-4957-85a1-36ae7eec2eac%2Fd24aa372-8503-40be-9818-3b63d3e66163%2F20b00ko_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Moments of the Region R
Consider the region R bounded by the function \( f(x) = 1 + \sin(x) \) and the x-axis, and the lines \( x = 1 \) and \( x = 4 \).
1. **Mass of the Region R:**
\[
\text{Mass} = M = \int_{1}^{4} (1 + \sin(x)) \, dx
\]
2. **Moment about y-axis:**
\[
M_y = \int_{1}^{4} x(1 + \sin(x)) \, dx
\]
3. **Moment about x-axis:**
\[
M_x = \int_{1}^{4} \frac{1 + \sin(x)}{2}(1 + \sin(x)) \, dx
\]
4. **Center of Mass:**
To find the center of mass, you need to calculate:
\[
\bar{x} = \frac{M_y}{M}
\]
\[
\bar{y} = \frac{M_x}{M}
\]
Make sure to round your final answers to two decimal places.
### Steps to Follow
- **Sketch the Region:**
Sketch the region defined by \( f(x) = 1 + \sin(x) \), the x-axis, and the vertical lines \( x = 1 \) and \( x = 4 \). Label the strip used for integration and the center of the strip.
- **Write the Definite Integrals:**
Set up the definite integrals for mass, and the moments about the x-axis and y-axis.
- **Evaluate the Integrals:**
Solve the definite integrals to find the moments and hence the center of mass (centroid) of the region R.
Use these steps to ensure a detailed understanding and accurate calculation of the moments and centroid.
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