Arc Length with Polar Coordinates — Question 7

Question 7

Problem

Find the length of $r=e^\theta$, $0\le\theta\le\ln2$.

Question 7 – 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 $r'=r$, speed is $\sqrt2e^\theta$.

4. Thus $L=\sqrt2[e^\theta]_0^{\ln2}=\boxed{\sqrt2}.$