Arc Length with Polar Coordinates — Question 5

Question 5

Problem

A circle of the same radius is $r=2R\cos\theta$. Which interval traces it once, and what length results?

Question 5 – Solution

Solution

1. Differentiate the polar radius to obtain $r'=dr/d\theta$ and choose an interval that traces the requested arc exactly once.

2. Use the polar arc-length formula $$L=\int_a^b\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta.$$ Simplify the expression under the square root before evaluating or reporting the integral.

3. $-\pi/2\le\theta\le\pi/2$ traces it once.

4. The speed simplifies to $2R$, so $L=2R\pi=\boxed{2\pi R}.$