Find the centroid of the lamina bounded by the x- and y-axes, the line x = 2, and the graph of y = e. Assume the density 8(x, y) = 1. (Use symbolic notation and fractions where needed. Give your answer as point coordinates in the form (*,*).) (x, y) =
Find the centroid of the lamina bounded by the x- and y-axes, the line x = 2, and the graph of y = e. Assume the density 8(x, y) = 1. (Use symbolic notation and fractions where needed. Give your answer as point coordinates in the form (*,*).) (x, y) =
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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![### Problem Statement
Find the centroid of the lamina bounded by the x-axis, y-axis, the line \( x = 2 \), and the graph of \( y = e^{-x} \). Assume the density \( \delta(x, y) = 1 \).
(Use symbolic notation and fractions where needed. Give your answer as point coordinates in the form \((*,*)\).)
\[
(\bar{x}, \bar{y}) = \text{[Input box]}
\]
### Explanation
This problem involves calculating the centroid of a region in the xy-plane. The region is bounded by:
- The x-axis (where \( y = 0 \))
- The y-axis (where \( x = 0 \))
- The vertical line \( x = 2 \)
- The curve of the exponential function \( y = e^{-x} \)
Given that the density \( \delta(x, y) \) is constant and equal to 1, the problem simplifies to finding the centroid of the described region without any additional weighting due to density variations.
### Approach
1. **Identify the Region**:
- Vertical boundaries are from \( x = 0 \) to \( x = 2 \).
- The top boundary is \( y = e^{-x} \).
- The bottom boundary is \( y = 0 \).
2. **Calculate Area (A)**:
- The centroid \((\bar{x}, \bar{y})\) formula requires the area of the region \((A)\), which is given by the definite integral of the top boundary minus the bottom boundary over the interval of \( x \), i.e.,
\[
A = \int_{0}^{2} e^{-x} \, dx
\]
3. **Calculate the Centroid Coordinates (\(\bar{x}, \bar{y}\))**:
- The x-coordinate of the centroid:
\[
\bar{x} = \frac{1}{A} \int_{0}^{2} x \cdot e^{-x} \, dx
\]
- The y-coordinate of the centroid:
\[
\bar{y} = \frac{1}{A} \int_{0}^{2} \frac{1}{2} (e^{-x})^2 \, dx
\]
These steps will yield the centroid's coordinates in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77548912-c51c-4c9d-8b51-f3905a3bec75%2F39497cff-02e6-4ccb-8ce5-43d054894a18%2Fa21c7du_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the centroid of the lamina bounded by the x-axis, y-axis, the line \( x = 2 \), and the graph of \( y = e^{-x} \). Assume the density \( \delta(x, y) = 1 \).
(Use symbolic notation and fractions where needed. Give your answer as point coordinates in the form \((*,*)\).)
\[
(\bar{x}, \bar{y}) = \text{[Input box]}
\]
### Explanation
This problem involves calculating the centroid of a region in the xy-plane. The region is bounded by:
- The x-axis (where \( y = 0 \))
- The y-axis (where \( x = 0 \))
- The vertical line \( x = 2 \)
- The curve of the exponential function \( y = e^{-x} \)
Given that the density \( \delta(x, y) \) is constant and equal to 1, the problem simplifies to finding the centroid of the described region without any additional weighting due to density variations.
### Approach
1. **Identify the Region**:
- Vertical boundaries are from \( x = 0 \) to \( x = 2 \).
- The top boundary is \( y = e^{-x} \).
- The bottom boundary is \( y = 0 \).
2. **Calculate Area (A)**:
- The centroid \((\bar{x}, \bar{y})\) formula requires the area of the region \((A)\), which is given by the definite integral of the top boundary minus the bottom boundary over the interval of \( x \), i.e.,
\[
A = \int_{0}^{2} e^{-x} \, dx
\]
3. **Calculate the Centroid Coordinates (\(\bar{x}, \bar{y}\))**:
- The x-coordinate of the centroid:
\[
\bar{x} = \frac{1}{A} \int_{0}^{2} x \cdot e^{-x} \, dx
\]
- The y-coordinate of the centroid:
\[
\bar{y} = \frac{1}{A} \int_{0}^{2} \frac{1}{2} (e^{-x})^2 \, dx
\]
These steps will yield the centroid's coordinates in
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