### Transcription and Explanation**Figure P5.42**This diagram illustrates a curve defined by the equation:\[ y = a \left(1 - \frac{x}{L} + \frac{x^2}{L^2}\right) \]**Explanation of the Diagram:**- **Axes:** The diagram is presented on a two-dimensional coordinate plane, with the horizontal axis labeled as \( x \) and the vertical axis labeled as \( y \). - **Curve:** The curve represents a quadratic function that starts at \( y = a \) when \( x = 0 \) and is defined over the interval from \( x = 0 \) to \( x = 2L \).- **Dimensions:** - The vertical line at \( y = a \) indicates the starting height of the curve when \( x = 0 \). - The curve is segmented into two equal parts along the \( x \)-axis, each of length \( L \).The curve starts at height \( a \) when \( x = 0 \), initially dips as \( x \) increases, and then gradually rises as \( x \) approaches \( 2L \). This kind of curve can be useful for understanding quadratic relationships and the effect of different variables on the shape of a function. ### Problem 5.42Determine by direct integration the centroid of the area shown.Determine by direct integration the centroid of the area shown. Express your answer in terms of \( a \) and \( b \).

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Answered: Determine by direct integration the…

Engineering

Civil Engineering

Determine by direct integration the centroid of the area shown.

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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Concept explainers

Coordinate Geometry of Survey

Coordinate geometry (COGO) is a method of inserting surveying or engineering data into geographic information system (GIS), computer-aided design (CAD), or mapping software. These data can be collected on the proposed construction site using standard surveying techniques and instrumentation or may be taken from existing maps, engineering plans, drawings, or records. Data gathered on the site includes coordinates of points, bearings, and distances between points. The data are noticed and noted through conventional surveying procedures, like a traverse or layout measurement series and it can be either digital or non-digital. The maps and plans are inclusive of boundaries, divisions, and other components, and these features are shown with coordinate values, the lengths, offset distance, and bearings between them.

Question

### Transcription and Explanation

**Figure P5.42**

This diagram illustrates a curve defined by the equation:

\[ y = a \left(1 - \frac{x}{L} + \frac{x^2}{L^2}\right) \]

**Explanation of the Diagram:**

- **Axes:** The diagram is presented on a two-dimensional coordinate plane, with the horizontal axis labeled as \( x \) and the vertical axis labeled as \( y \).

- **Curve:** The curve represents a quadratic function that starts at \( y = a \) when \( x = 0 \) and is defined over the interval from \( x = 0 \) to \( x = 2L \).

- **Dimensions:**
- The vertical line at \( y = a \) indicates the starting height of the curve when \( x = 0 \).
- The curve is segmented into two equal parts along the \( x \)-axis, each of length \( L \).

The curve starts at height \( a \) when \( x = 0 \), initially dips as \( x \) increases, and then gradually rises as \( x \) approaches \( 2L \). This kind of curve can be useful for understanding quadratic relationships and the effect of different variables on the shape of a function.

### Problem 5.42

Determine by direct integration the centroid of the area shown.

Determine by direct integration the centroid of the area shown. Express your answer in terms of \( a \) and \( b \).

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