Mechanics Homework 20-4 Name_______________ Mr. Haynes Date_______________

Problem 20-4 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:

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

$$A = \pi r^2$$, $$\bar{y} = \frac{4 R}{3 \pi}$$.