Arc Length with Polar Coordinates — Question 6

Question 6 – 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. Here $r>0$ and $r'=2\cos20\theta$. Thus $\sqrt{r^2+(r')^2}\ge r$, with strict inequality except at finitely many angles. Consequently $L>\int_0^{2\pi}(1+0.1\sin20\theta)\,d\theta=2\pi$, the unit circle’s length.