Arc Length with Polar Coordinates — Question 1

Question 1

Problem

Find the length of the cardioid $r=1+\cos\theta$.

Question 1 – 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. The speed simplifies to $2|\cos(\theta/2)|$.

4. Integrating from $0$ to $2\pi$ gives $\boxed{8}$.