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Chapter 8: Problem 34

Find the centroid of the region. $$ \text { Semicircle of radius } r \text { with center at the origin } $$

Short Answer

Expert verified
The centroid of the semicircle of radius \r\ with center at the origin is \left(0, \frac{4r}{3\pi}\right)\.

Step by step solution

01

Understanding the Centroid

The centroid of a shape is the average position of all the points in the shape. For a semicircle, symmetry helps simplify the problem.
02

Setting Up the Coordinates

Place the semicircle in a coordinate system such that the center of the semicircle is at the origin \(0, 0\) and the flat edge lies on the x-axis. This semicircle has radius r.
03

Coordinates of the Centroid

The centroid (geometric center) of a semicircle lies on the y-axis due to symmetry. Thus, the x-coordinate of the centroid \(\bar{x}\) is 0. We need to find the y-coordinate \(\bar{y}\).
04

Using the Formula for the y-coordinate

For a semicircle with its diameter along the x-axis, with center at the origin and radius r, the formula for the y-coordinate of the centroid is: \(\bar{y} = \frac{4r}{3\pi}\).
05

Calculating the y-coordinate

Substitute \r\ (radius of the semicircle) into the formula to get the final result for the y-coordinate: \(\bar{y} = \frac{4r}{3\pi}\).
06

Conclusion

Thus, the coordinates of the centroid of the semicircle are \(0, \frac{4r}{3\pi}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

coordinate system
A coordinate system is essential for locating points in a plane. It's like a grid where we can define the positions of objects. Imagine a giant graph paper.
The most common coordinate system is the Cartesian coordinate system, which uses an x-axis (horizontal) and a y-axis (vertical). When we place a semicircle in this grid:
  • We position the center at the origin (0,0).
  • The flat edge of the semicircle lies on the x-axis.
  • The radius of the semicircle is denoted by r.
This setup helps us conveniently use formulas and calculate the centroid.
centroid formula
The centroid of a shape is its geometric center or average location of its points. For regular shapes, like circles and triangles, there are specific formulas to find the centroid.
For a semicircle, things are a bit special. The formula for the y-coordinate of the centroid of a semicircle (with its flat edge on the x-axis) is: \(\bar{y} = \frac{4r}{3\pi}\) where r is the radius. The x-coordinate is simple, as it lies on the y-axis, making \(\bar{x} = 0\).
By substituting the radius into the y-coordinate formula, you get the exact location of the centroid.
symmetry in geometry
Symmetry plays a vital role in geometry, especially when finding centroids. A shape is symmetrical if one half is a mirror image of the other half.
  • For our semicircle, there is vertical symmetry along the y-axis.
  • This means the x-coordinates of all points balance out to zero.
Hence, the x-coordinate \(\bar{x}\) of the centroid is 0.
Symmetry simplifies calculations, allowing us to focus on the y-coordinate for the centroid.

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Most popular questions from this chapter

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