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Use double integrals to find the center of mass of the triangular plate in the first quadrant bounded by x+ y = 5 with variable density 8(x, y) = 5x 1. Sketch region. Find the mass of the plate. 4- 3- 2- 1- 2 1 0 1 -4 -3 2 3 2- 3 4- 2. Find the moment about the y-axis, M, and the moment about the x-axis, M, 3. The center of mass is (x,J)=

Use double integrals to find the center of mass of the triangular plate in the first quadrant bounded by x+ y = 5 with variable density 8(x, y) = 5x 1. Sketch region. Find the mass of the plate. 4- 3- 2- 1- 2 1 0 1 -4 -3 2 3 2- 3 4- 2. Find the moment about the y-axis, M, and the moment about the x-axis, M, 3. The center of mass is (x,J)=

Calculus: Early Transcendentals
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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**Topic: Calculating the Center of Mass of a Triangular Plate Using Double Integrals** Use double integrals to find the center of mass of the triangular plate in the **first quadrant** bounded by the equation \( x + y = 5 \) with a variable density function \( \delta(x, y) = 5x \). **1. Sketch the Region and Find the Mass of the Plate:** A graph is provided with the \(x\)-axis and \(y\)-axis both ranging from \(-5\) to \(5\). You need to sketch the triangular region bounded by the line \(x + y = 5\), focusing on the first quadrant where \(x \geq 0\) and \(y \geq 0\). **2. Calculate the Moments:** - **Moment about the \(y\)-axis (\(M_y\))**: This moment accounts for the distribution of mass relative to the \(y\)-axis. - **Moment about the \(x\)-axis (\(M_x\))**: This moment takes into account the distribution of mass relative to the \(x\)-axis. **3. Determine the Center of Mass \((\bar{x}, \bar{y})\):** Here, you need to find the coordinates of the center of mass using the moments calculated in the previous step. **Graph Explanation:** The grid is a coordinate plane showing positive and negative divisions on both the \(x\) and \(y\) axes, with intervals marked from \(-5\) to \(5\). The line \(x + y = 5\) can be drawn on this grid, and you'll focus on the section of this line that lies within the first quadrant. This section, along with the axes, forms a triangular region where the calculations will be applied.
Transcribed Image Text:**Topic: Calculating the Center of Mass of a Triangular Plate Using Double Integrals** Use double integrals to find the center of mass of the triangular plate in the **first quadrant** bounded by the equation \( x + y = 5 \) with a variable density function \( \delta(x, y) = 5x \). **1. Sketch the Region and Find the Mass of the Plate:** A graph is provided with the \(x\)-axis and \(y\)-axis both ranging from \(-5\) to \(5\). You need to sketch the triangular region bounded by the line \(x + y = 5\), focusing on the first quadrant where \(x \geq 0\) and \(y \geq 0\). **2. Calculate the Moments:** - **Moment about the \(y\)-axis (\(M_y\))**: This moment accounts for the distribution of mass relative to the \(y\)-axis. - **Moment about the \(x\)-axis (\(M_x\))**: This moment takes into account the distribution of mass relative to the \(x\)-axis. **3. Determine the Center of Mass \((\bar{x}, \bar{y})\):** Here, you need to find the coordinates of the center of mass using the moments calculated in the previous step. **Graph Explanation:** The grid is a coordinate plane showing positive and negative divisions on both the \(x\) and \(y\) axes, with intervals marked from \(-5\) to \(5\). The line \(x + y = 5\) can be drawn on this grid, and you'll focus on the section of this line that lies within the first quadrant. This section, along with the axes, forms a triangular region where the calculations will be applied.
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