h y= ) Locate the centroid (x, y) of the shaded area shown. y -X
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![### Locating the Centroid of a Shaded Area
This section will guide you through locating the centroid (\(\bar{x}, \bar{y}\)) of a shaded area defined by a specific function.
#### Graph Description:
1. **Axes**:
- The graph is plotted in the Cartesian coordinate system with the horizontal axis as \(x\) and the vertical axis as \(y\).
2. **Curve and Shaded Region**:
- The shaded region is bounded by the curve \(y = -\frac{h}{a^4} x^4\), the y-axis, the x-axis, and a vertical line at \(x = a\).
- The shaded region starts at the origin (0,0), extends to the point where \(x = a\), and the curve dips downwards forming a concave shape.
3. **Parameters in the Equation**:
- \(h\): Represents the height of the shaded area along the y-axis.
- \(a\): Represents the width of the shaded area along the x-axis.
#### Objective:
To locate the centroid (\(\bar{x}, \bar{y}\)) of the given shaded area.
**\( \bar{x} \)** and **\( \bar{y} \)** represent the coordinates of the centroid, which is the theoretical balancing point of the area.
### Equations and Coordinates:
1. **Equation of the Bounding Curve**:
\[
y = -\frac{h}{a^4} x^4
\]
In the next steps, we will apply integration techniques specific to centroid calculations to find the coordinates \(\bar{x}\) and \(\bar{y}\).
### Steps Overview:
1. **Find the Area (A)**: Integrate the function over the given boundaries.
2. **Find \( \bar{x} \)**: Using the formula:
\[
\bar{x} = \frac{1}{A} \int (x \cdot dA)
\]
3. **Find \( \bar{y} \)**: Using the formula:
\[
\bar{y} = \frac{1}{A} \int (y \cdot dA)
\]
Further detailed calculations and examples will be provided in the following sections.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbad2734e-649a-4533-b9cc-3c645832697e%2F0f22f17c-8426-484a-ab9d-6d075af96260%2F5ryew3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Locating the Centroid of a Shaded Area
This section will guide you through locating the centroid (\(\bar{x}, \bar{y}\)) of a shaded area defined by a specific function.
#### Graph Description:
1. **Axes**:
- The graph is plotted in the Cartesian coordinate system with the horizontal axis as \(x\) and the vertical axis as \(y\).
2. **Curve and Shaded Region**:
- The shaded region is bounded by the curve \(y = -\frac{h}{a^4} x^4\), the y-axis, the x-axis, and a vertical line at \(x = a\).
- The shaded region starts at the origin (0,0), extends to the point where \(x = a\), and the curve dips downwards forming a concave shape.
3. **Parameters in the Equation**:
- \(h\): Represents the height of the shaded area along the y-axis.
- \(a\): Represents the width of the shaded area along the x-axis.
#### Objective:
To locate the centroid (\(\bar{x}, \bar{y}\)) of the given shaded area.
**\( \bar{x} \)** and **\( \bar{y} \)** represent the coordinates of the centroid, which is the theoretical balancing point of the area.
### Equations and Coordinates:
1. **Equation of the Bounding Curve**:
\[
y = -\frac{h}{a^4} x^4
\]
In the next steps, we will apply integration techniques specific to centroid calculations to find the coordinates \(\bar{x}\) and \(\bar{y}\).
### Steps Overview:
1. **Find the Area (A)**: Integrate the function over the given boundaries.
2. **Find \( \bar{x} \)**: Using the formula:
\[
\bar{x} = \frac{1}{A} \int (x \cdot dA)
\]
3. **Find \( \bar{y} \)**: Using the formula:
\[
\bar{y} = \frac{1}{A} \int (y \cdot dA)
\]
Further detailed calculations and examples will be provided in the following sections.
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