Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles and using additivity of moments. y (x, y) = ([ T -3 -2 -1 1 2 3

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Finding the Centroid of a Composite Shape**

To find the centroid of the region shown without using integration, locate the centroids of the component rectangles and triangles, and apply the concept of additivity of moments.

### Diagram Description:
- **Axes:** The x-axis and y-axis are labeled with units marked from -4 to 3 on the x-axis and 0 to 3 on the y-axis.
- **Shape:** The shaded region is composed of a rectangle and a right triangle:
  - The **rectangle** extends from x = -4 to x = -2 and from y = 0 to y = 2.
  - The **right triangle** extends from x = -2 to x = 1, with its base on the x-axis and the top aligned at y = 3 at x = -2, sloping downwards to the x-axis at x = 1.

### Task:
- **Objective:** Find the centroid \((\bar{x}, \bar{y})\) by using the centroids of these simpler shapes and calculating the weighted average based on their areas.

\[ (\bar{x}, \bar{y}) = (\text{Enter the coordinates here}) \]

**Note:** To solve this, determine the areas and centroids of the individual shapes, then calculate the combined centroid using the formula for composite areas.
Transcribed Image Text:**Finding the Centroid of a Composite Shape** To find the centroid of the region shown without using integration, locate the centroids of the component rectangles and triangles, and apply the concept of additivity of moments. ### Diagram Description: - **Axes:** The x-axis and y-axis are labeled with units marked from -4 to 3 on the x-axis and 0 to 3 on the y-axis. - **Shape:** The shaded region is composed of a rectangle and a right triangle: - The **rectangle** extends from x = -4 to x = -2 and from y = 0 to y = 2. - The **right triangle** extends from x = -2 to x = 1, with its base on the x-axis and the top aligned at y = 3 at x = -2, sloping downwards to the x-axis at x = 1. ### Task: - **Objective:** Find the centroid \((\bar{x}, \bar{y})\) by using the centroids of these simpler shapes and calculating the weighted average based on their areas. \[ (\bar{x}, \bar{y}) = (\text{Enter the coordinates here}) \] **Note:** To solve this, determine the areas and centroids of the individual shapes, then calculate the combined centroid using the formula for composite areas.
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