Arc Length with Polar Coordinates — Question 9

Question 9

Problem

A polar path has constant arc-length speed $5$ and runs for an angular interval of width $2\pi/3$. What is its length?

Question 9 – 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. Arc length is the integral of speed: $L=5(2\pi/3)=\boxed{10\pi/3}.$