Arc Length with Polar Coordinates — Question 8

Question 8

Problem

Explain why integrating the rose $r=\cos3\theta$ from $0$ to $2\pi$ overcounts its geometric length.

Question 8 – 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. For odd $3$, the full curve is completed on $0\le\theta\le\pi$; the next $\pi$ retraces it because negative radius duplicates opposite points.

4. The $0$ to $2\pi$ integral counts every petal twice.