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Locate the centroid x¯ of the shaded area.

Locate the centroid x¯ of the shaded area.

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Locate the centroid x¯ of the shaded area.

**Consider the area shown in (Figure 1). Suppose that \( a = 9 \, \text{m} \), \( k = \frac{1}{9} \, \text{m}^{-1} \). **Figure Explanation:** The figure displays a shaded region in the first quadrant of the \(xy\)-plane. The area is bounded by the following components: - **Horizontal Axis (x-axis):** Extends from \(x = 0\) to \(x = a\). - **Vertical Axis (y-axis):** Extends from \(y = 0\) upwards. - **Vertical Line:** \(x = a\), parallel to the \(y\)-axis. - **Curve:** The curve is described by the equation \(y = kx^2\), and it starts at the origin \((0, 0)\) and curves upwards passing through the point \((a, a)\). The shaded region, which resembles a quarter of a parabola, is contained within these boundaries. The curve is a segment of the parabola described by the given equation, and the limits define a square area with side \(a\).
Transcribed Image Text:**Consider the area shown in (Figure 1). Suppose that \( a = 9 \, \text{m} \), \( k = \frac{1}{9} \, \text{m}^{-1} \). **Figure Explanation:** The figure displays a shaded region in the first quadrant of the \(xy\)-plane. The area is bounded by the following components: - **Horizontal Axis (x-axis):** Extends from \(x = 0\) to \(x = a\). - **Vertical Axis (y-axis):** Extends from \(y = 0\) upwards. - **Vertical Line:** \(x = a\), parallel to the \(y\)-axis. - **Curve:** The curve is described by the equation \(y = kx^2\), and it starts at the origin \((0, 0)\) and curves upwards passing through the point \((a, a)\). The shaded region, which resembles a quarter of a parabola, is contained within these boundaries. The curve is a segment of the parabola described by the given equation, and the limits define a square area with side \(a\).
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