Determine the x- and y-coordinates of the centroid of the shaded area. Assume a = 8 in., b = 10 in. y T a Answer: b x = ky² (x, y) = (i x i ) in.
Determine the x- and y-coordinates of the centroid of the shaded area. Assume a = 8 in., b = 10 in. y T a Answer: b x = ky² (x, y) = (i x i ) in.
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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Question
1
![**Determining the Centroid of a Shaded Area**
---
**Objective:** Determine the \( x \)- and \( y \)-coordinates of the centroid of the given shaded area.
**Given:**
- \( a = 8 \) in.
- \( b = 10 \) in.
**Description:**
The diagram illustrates a shaded region bounded by the y-axis on the left, a horizontal line segment at \( y = a \), and an upper curve defined by the equation \( x = ky^2 \). The rightmost boundary is a vertical line segment at \( x = b \), and the leftmost boundary intersects the y-axis.
**Visual Explanation:**
- The shaded region forms part of a parabola extending upwards, with an area bounded by a parabola.
- The parabola has the equation \( x = ky^2 \), indicating the relationship between \( x \) and \( y \).
- The shaded area spans from \( y = 0 \) to \( y = a \), horizontally from \( x = 0 \) to \( x = b \).
**Solution Format:**
Answer:
\[
(\overline{x}, \overline{y}) = (\boxed{\enspace \enspace i \enspace \enspace}, \boxed{\enspace \enspace i \enspace \enspace}) \text{ in.}
\]
To determine the centroid, calculate the integral components for \( x \) and \( y \) centroids.
**Note for Students:** Use appropriate integral formulas considering the boundaries and the shape.
This educational description and problem-solving methodology will guide you to find the centroid for similar problems involving parabolic areas and other shapes.
---
This structured approach ensures clarity and assists in understanding the integral concept of centroids for areas defined by curves.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc59a6d07-a993-40f7-b57e-cdb2ad079d78%2Fce4080d6-dcbd-4391-bc9d-53ad65e40934%2Fxzxpakq_processed.png&w=3840&q=75)
Transcribed Image Text:**Determining the Centroid of a Shaded Area**
---
**Objective:** Determine the \( x \)- and \( y \)-coordinates of the centroid of the given shaded area.
**Given:**
- \( a = 8 \) in.
- \( b = 10 \) in.
**Description:**
The diagram illustrates a shaded region bounded by the y-axis on the left, a horizontal line segment at \( y = a \), and an upper curve defined by the equation \( x = ky^2 \). The rightmost boundary is a vertical line segment at \( x = b \), and the leftmost boundary intersects the y-axis.
**Visual Explanation:**
- The shaded region forms part of a parabola extending upwards, with an area bounded by a parabola.
- The parabola has the equation \( x = ky^2 \), indicating the relationship between \( x \) and \( y \).
- The shaded area spans from \( y = 0 \) to \( y = a \), horizontally from \( x = 0 \) to \( x = b \).
**Solution Format:**
Answer:
\[
(\overline{x}, \overline{y}) = (\boxed{\enspace \enspace i \enspace \enspace}, \boxed{\enspace \enspace i \enspace \enspace}) \text{ in.}
\]
To determine the centroid, calculate the integral components for \( x \) and \( y \) centroids.
**Note for Students:** Use appropriate integral formulas considering the boundaries and the shape.
This educational description and problem-solving methodology will guide you to find the centroid for similar problems involving parabolic areas and other shapes.
---
This structured approach ensures clarity and assists in understanding the integral concept of centroids for areas defined by curves.
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