Mechanics Homework | 21-4 1024 - Centroid of a semi-circle by integration using polar coordinates | Name_______________ |

Mr. Haynes | Date_______________ |

**Problem 21-4** First, watch this Khan Academy Video where Sal discusses how to integrate using polar coordinates. In particular, he shows that the differential element of area in polar coordinates is \(dA = \frac{1}{2} r^2\, d\theta \). Then prove, using integration with polar coordinates, that the area of a semi-circle is \(\pi r^2\) and that \(\bar{y} = \frac{4 r}{3 \pi}\).

Hints:

- The equation of a circle with radius \(r\) in polar coordinates is \(r(\theta) = r\).
- The process to find area and centroids is exactly the same as previous problems, only the form of the integrals is different.
- Finding the area of a circle using polar coordinates is trivially easy.
- To find \(Q_x\) you will need to determine \(\bar{y}_{el}\) as a function of \(r\) and \(\theta\).