Arc Length with Polar Coordinates — Question 4

Question 4

Problem

Use the polar arc-length formula to recover the circumference of $r=R$.

Question 4 – 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. Since $r'=0$, speed is $R$.

4. Hence $L=\int_0^{2\pi}R\,d\theta=\boxed{2\pi R}.$