Arc Length with Polar Coordinates — Question 2

Question 2

Problem

Find the length of one petal of $r=3\cos2\theta$.

Question 2 – 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. On $[-\pi/4,\pi/4]$, speed is $3\sqrt{\cos^22\theta+4\sin^22\theta}$.

4. Thus $\boxed{L=3\int_{-\pi/4}^{\pi/4}\sqrt{1+3\sin^22\theta}\,d\theta}$; it is an elliptic integral.