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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
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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**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.
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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